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PRACTICAL ASTRONOMY 



BY 

P' S. MICHIE 

AND 

F. 8. pARLOW 



/ 



^^- 



O 



JlJr! 26 



""^^ 



U. S. M. A. PkE.S^ 




AND lilNDERY. 



1891. 



'*^h^ 



Copyright, 1891, 

BY 
P. S. MiCHIE. 



PREFACE 



1 



o supplement the course in Descriptive Astronomy, now studied 
from Professor C. A. YoUNG^'s excellent text-book of General 
Astronomy, the following Course of Practical Astronomy has been 
prepared for the use of the Cadets of the U. S. Military Academy, That 
jiortion relating to the forms and text for practical problems was pre- 
pared by Lieutenant Frank S. Harlow, U. S. Artillery, at present in 
charge of the West Point Observatory. Owing to the limited time 
available for this portion of the Course in the Department of Natural 
Philosophy it was thought best to limit the application to but a few of 
the more important problems of time, latitude, longitude, etc., and to 
make these few as thorough as possible. The American Ephemeris and 
Nautical Almanac is to be freely used in the solution of these problems, 
and the required data in each case is to be obtained by careful observa- 
tions either at the field or the permanent observatory. Subsequently 
llie Cadets are taught to use the principal portable astronomical in- 
struments during the Summer Encampment. The Course of Astronomy 
will then comprise: 

1. A brief exposition of those principles of Physical Astronomy 
considered as a branch of Mechanics, such as is found in Mtchie's 
Mechanics. 
^. The Descriptive Greneral Astronomy comprised in Young's text- 
book. 
l\. The Practical course herewith presented ; and 
4. The use of portable astronomical instruments and the permanent 
instruments of the fixed observatory, as far a* opportunity and 
time will permit, during the encampment. 

P. S. M. 
West Point, February, 1891. 



PRACTICAL ASTRONOMY. 



PRACTICAL ASTRO^^OMY. 



EPHEMERIS. 

Ephemeris. — The numerical values of the coordinates 
of the principal celestial hodies. together with the elements 
of position of the circles of reference, are recorded for given 
equi-distant instants of time in an Astronomical Ephemeris. 

The "American Ephemeris and Nautical Almanac" is 
published by the United States Government, generally three 
years in advance of the year of its title, and comprises three 
parts, viz.: 

Part I. — Ephemeris for the Pleridian of Greenwich, which 
gives the heliocentric and geocentric positions of the major 
planets, the ephemeris of the sun, and other fundamental 
astronomical data for equi-distant intervals of mean Green- 
wich time. 

Part II. — Ephemeris for the Meridian of Washington, which 
gives the ephemerides of the fixed stars, sun, moon and 
major planets, for transit over the meridian of Washington, 
and also the mean places of the fixed stars, with the data 
for their reduction. 

Part III.— Phenomena, which contains prediction of phe- 
nomena to be observed, with data for their computation. 

EPHEMERIS OF THE SUN. 

To construct the ephemeris of the sun it is necessary to 
compute its tables: these are 

1. The table of Epochs. 

2. The table of Longitudes of Perigee. 

3. The table of Equations of the Center, and its corrections. 

4. The table of the Equations of the Equinoxes in Longitude. 



PRACTICAL ASTRONOMY. 



In Mechanics * it was shown that the Earth's undisturbed 
orbit is an ellipse, having one of its foci at the sun's center, 
and that the earth's angular velocity is 

dO' h 



dt r^' 



(551) 



its radius vector. 



a{l-e') 



r = - — ^ ^ ; (610) 

1 + e cos ^ ' ^ ^ 



its constant sectoral area, 

Ifi = V/7a{l-^ : (615) 

and its periodic time. 

In these expressions 0' is the angle made by the earth's 
radius vector with any assumed right line drawn through 
the sun's center, 6 that included between the radius vector 
and the line of apsides estimated from perihelion, and n is 
the mean motion of the earth in its orbit. 

From (551), (615) and (616), we have 



de^ _dl__ V^'a {1 -e') (1 fe cos 0)' 
dt~dt~ a" (1 -^ 



/ 



y_ (l_+_ecos60' _ (1 -f e cos d f 

and therefore 

ndt=(l — e')i (1 + e cos 0)-^ d 6'. (2) 

Since e varies but little from 0.01 678 (see Art. 185, Youngf), 
we may omit all terms containing the third and higher 
powers of e in the development of the second member of the 

preceding equation. Then, after substituting — for 

cos^^, we have 

ndt = dd' -2e cos Ode + ^6^ cos 2ed{2 6) + etc. (3) 

Integrating we have 

nt+ C=e'-2esm (9 + f e^ sin 2 6> + etc. (4) 



* Michie's Mechanics, 4th Edition, 
f Young's General Astronomy. 



PRACTICAL ASTRONOMY. 7 

Let the line from which 6' is estimated be that drawn 
through the sun and the true vernal equinox of the date: 
then when 6 is zero, 6' will be the true longitude of peri- 
helion; let this be represented by Ip and the time of peri- 
helion passage by tp-, then from Eq. (4), we have 

ntp+C=lp. (5) 

Subtracting this from Eq. (4), we have 

n{t-tp) = 6' -lp-2e sin 6* + f e^ gin 2 0, (6) 

which since 

d'-ip^d. (7) 

reduces to 

n {t - tp) = (6' -lp)-2e sin {6' - y + | e' sin 2 {d' - Q, (8) 
in which the first member, being the angular distance of 
the mean place of the earth from perihelion, is the mean 
anomaly. Transposing Ip we have 

n{t- tp) + lp= L = 0' -2 e sin {6' - Ip) + j c^sin 2 (6' - Q, (9) 
in which Im is the true longitude of the mean place of the 
earth in its orbit. 

Let L be the true longitude of the earth's mean place at 
any epoch, and t be any interval of time after this epoch: 
then will 

U, = L + n t, (10) 

and we have 

L + nt^B' -2e^\\i (6*' - y + f e' sin 2 (O'-lp). (11) 

To find the values of the four unknown quantities, L, n, e, 
and Ip, take four observations of right ascension and decli- 
nation of the sun at different times, and (Art. 180) find the 
corresponding true longitudes of the sun. Since the longi- 
tudes of the sun and earth always differ by 180°, the addition 
of 180° to the deduced longitudes of the sun will give the 
values of 0' corresponding to the values of t in the follow- 
ing equations. 

L-hnt,= 6,' -2e sin (6*/ - Q, ] 
L-\-ut2= 0-/ - 2 e sin {6/ - Q, 
L + nt3 = ds' — 2e sin (0/-lp), 
L-\-nt^ = 6,' -2e sin ((9/ - Ip). 



(12) 



8 PRACTICAL ASTRONOMY. 

These observations repeated at different times will deter- 
mine the changes that take place in n, e, and l^-, from the 
last two the variations in the eccentricity and the rate of 
motion of perihelion can be found. 

Having in this manner found the elements of the earth's 
place and motion, the corresponding true longitude of the 
sun at any instant can be obtained by adding to that of the 
earth 180°. The first member of Eq. (11) plus 180° will then 
give for any instant the true longitude of the sun's mean 
place. The difference between the longitudes of the sun's 
true and mean places at any instant is the Equation of the 
Center for that instant. 

From the preceding elements let it be required to construct 
the Ephemeris of the Sun. 

1 . The Table of Epochs. — Take mean midnight, December 
31 — January 1, 1890, as the epoch. To the true longitude of 
the sun's mean place at that epoch, add the product of the 
sun's mean motion n, by the number of mean solar days 
after the epoch, subtracting 360° when this sum is greater 
than 360°. These longitudes with their corresponding times 
being tabulated, form the table of epochs, from which the 
true longitude of the mean place of the sun can be found by 
inspection for any day, hour, minute or second. 

2. The Table of Longitudes of Perigee. — The longitude of 
perihelion increased by 180° is the corresponding longitude 
of perigee. Hence the former being found, and its rate of 
change determined, the addition of 180° to each longitude of 
perihelion will give the longitude of perigee, and these values 
being tabulated form the table of longitudes of perigee. 

3. The Table of Equations of the Center.— The difference 
between the true and mean anomalies at any instant, given 
by the first of Eqs. (650), Mechanics, 

— nt — 'Ze sin nt -\- ^e^ sin %nt -{- etc., (13) 

is called the Equation of the Center, and is known when 
n and e are known. 

Assuming e to be constant and causing nt to vary from 
0° to 360°, the resulting values of the second member of the 



PRACTICAL ASTRONOMY. I) 

equation will form a table of the equations of the center. 
The errors in these values arise from the small variations in 
the values of e; these errors can be found by substituting 
in the sejcond member of the above equation the actual 
values of e at the time and the differences being tabulated 
will give a table by which the equations of the center may 
be corrected from time to time. 

4. The Perturbations in Longitude of the earth arising 
from the attractions of the planets (especially Venus and 
Jupiter), are the same for the sun; these are computed by 
the methods indicated in Physical Astronomy, (see Art. 174, 
Mechanics.) and then tabulated. 

5. The Sun's Aberration is taken to be constant, amount- 
ing to — 2()"."<J5 and is included in the table of epochs. 

Ephemeris of the Sun.— The above tables having been 
computed, we proceed as follows: 

1. From the table of epochs take out the true longitude of 
the sun's mean place corresponding to the exact instant 
considered. 

•I. From the table of longitudes of perigee take the true 
longitude of perigee; the difference between this and the 
true longitude of the sun's mean place is the mean anomaly. 

3. With the mean anomaly as an argument find the corre- 
sponding value of the equation of the center from its table, 
and add it with its proper sign to the true longitude of the 
sun's mean place; the result will be the true longitude of the 
sun's true place; hence the 

Sun's true longitude = True longitude of sun's mean place 
± Equation of center ± Perturbations in longitude ± Cor- 
rections to pass from the true equinox of epoch to true equi- 
nox of date. These latter corrections are due to Precession 
and Nutation and will be explained further on. 

•4. Having the true longitude of the sun and the obliquity 
of the ecliptic, the corresponding Right Ascension and De- 
clination of the sun can be computed for the same instant 
by the method explained in Art. 180, Astronomy. 

6. Earth's Radius Vector. — Substituting the values of e 
and n t, in the second of Eqs. (650), Mechanics, will give 



10 PRACTICAL ASTRONOMY. 

the values of the distance of the sun from the earth in terms 
of the mean distance a\ thus 

/ e^ 3 e^ 

r = a M — e cos nt + - (1 — cos 2 n t) — — (cos 3 n t ~ cos ii t) 



+ etc. ). (14) 

7. The Sun's Horizontal Parallax.— Knowing r, the radius 
vector, p the earth's radius at the observer's place, and oo 
the number of seconds in radius = 206264". 8, the value of 
the sun's horizontal parallax P, can be found from 

in which r is given in terms of p. 

8. The Sun's Apparent Diameter, is found from 

Pd , ; 

^=~fr^ (16) 

in which s is the apparent diameter in angular measure, 
and d, the real diameter in the same units in which p is 
given. 

9. Equation of the Equinoxes in Longitude.— Owing to 

precession and nutation the true equinox does not move with 
uniform motion. Due to precession, the true equinox moves 
with a retrograde motion of about 50". 2 in a year; and due 
to nutation the true pole describes a little ellipse on the ce- 
lestial sphere whose longer axis is directed to the pole of the 
ecliptic and is about 18". 5 in angular measure while its con- 
jugate axis is about 13". 74. The mean equinox and the true 
equinox are both on the ecliptic and their distance apart in 
angular measure is called the Equation of Equinoxes in 
Longitude; its projection on the celestial equator is called 
the Equation of Equinoxes in Right Ascension; the inter- 
section of the declination circle, which projects the mean 
equinox, with the equator is called the Reduced Place of the 
Mean Equinox. 

The Equation of Equinoxes in Longitude is a function of 
the longitude of the moon's node, the longitude of the sun, 
and the obliquity of the ecliptic. Separate tables are con- 



PRACTICAL ASTRONOMY. 11 

structed for this correction, in which the arguments for enter- 
ing them are the obliquity and longitude of the moons node. 
and the obliquity and the longitude of the sun; the sum of 
the two corrections is the value of the equation of the equi- 
noxes in longitude at the corresponding times. 

10. Equation of Time. — If. at the instant when the true 
sun's mean place coincides with the mean equinox, an im- 
aginary point should leave the reduced place of the mean 
equinox and travel with uniform motion on the celestial 
equator, returning to its'starting point at the instant the true 
sun's mean place next again coincides with the mean equi- 
nox, such a point is called a MecDi Sun. Time measured by 
the hour angles of this point is called Mean Solar Time. 
The angle included between the declination circles passing 
through the center of the true sun and this point at any in- 
stant is called the Equation of Time for that instant; its 
value, at any instant, added algebraically to mean or appar- 
ent solar time will give the other. As the apparent time can 
be found by direct observation the equation of time is usually 
employed as a correction to pass from apparent to mean solar 
time. Since the mean sun moves in the celestial equator 
with the mean motion of the true sun in longitude, we have 
for the value of the equation of time. 

E = Mean Longitude of the true sun ± Equation of the 
Equinoxes in Right Ascension— the Right Ascension 
of the true sun. (17) 

11. Referring to the American Ephemeris of 1890, we see 
that Page I of each month contains the Sun's Apparent R. A. , 
Declination, Semi-diameter, Sidereal time of semi-diameter 
passing the meridian, at Greenwich apparent noon, together 
with the values for their respective hourly changes: the 
latter being computed from the values of their differential 
co-efficients. From these we can find the corresponding data 
for any other meridian. Page II contains similar data for 
the epoch of Greenwich mean noon, and in addition the 
sidereal time or R. A. of the mean sun. Page III contains 
the sun's true longitude and latitude, the logarithm of the 
earth's radius vector and the mean time of sidereal noon. 



12 PRACTICAL ASTRONOMY. 

The obliquity, precession, and sun's mean horizontal parallax 
for the year, are found on page 278 of the Ephemeris. All 
these constitute an Ephemeris of the Sun. 

From the hourly changes the elements for any meridian 
can be readily computed. 

THE EPHEMERIS OF THE MOON. 

The Ephemeris of the Moon consists of tables giving 
the Moon's Right Ascension and Declination for every hour 
of Greenwich mean time, with the changes for each minute; 
the Apparent Semi-diameter, Horizontal Parallax, Time of 
upper transit on the Greenwich Meridian, and Moon's Age. 
In order to compute these, it is first necessary to find the 
True Longitude of the Moon, its True Latitude, the Longitude 
of the Moon's Node, the Inclination of the Moon's Orbit to 
the Ecliptic, and the Longitude of Perigee. 

1. The Elements c f the Lunar Orbit.— Let DC he the 

intersection of the celestial sphere by the plane of the lunar 
orbit; VB the ecliptic, and VA the equinoctial; I^ the 
vernal equinox, N the ascending node, P the perigee, and 
il/i, Jfs. J^3. Mi, the geocentric places of the moon's center at 
four consecutive times fi, tz, ts. ti. These moon places are 
given by observed right ascensions and declinations taken 
at some observatory and corrected first for refraction to get 
the true altitude; second, for semi-diameter to change the 
observations from the limb of the moon to its center; and 
third, for parallax to refer the observation to the earth's 
center. These reduced right ascensions and declinations are 
then converted into the corresponding celestial longitudes 
and latitudes by the method described in Art. 180, Astronomy. 
Referring to the figure, assume the following notation: 

r = VN, the longitude of the node; 
/ = C NB, the inclination of the orbit; 
/, ^ rOu the longitude of M,\ 
J 2 — I Og, the longitude of Mo\ 
} J = M^ Oi, the latitude of M^\ 
?2 = M^ O2, the latitude of i)/^; 



PRACTICAL ASTRONOMY. 



13 



^•, = rEX+ XEM,, the orbit longitude of 31,-, 

J) = 1^ E X -^ XEP, the orbit longitude of perigee: 

cp = PEMi = i\ —p, the true anomaly of il/,; 

e = eccentricity of orbit; 

}u = mean motion of moon in its orbit: 

fi = time since epoch for M^\ 

L — mean orbit longitude at epoch. 

To find y and /. we have from the right-angled spherical 
triangles il/, .Vj O^ and M., X^ O.,, 

sin (l^~ v) — cot / tan Ai \ 
sin (1-2 — y) = cot / tan ^o ) 

and by division. 

sin (/, — ^) _ tan A^ 
sin {k — ^) ~ tan A^ ' 

Adding unity to both members, reducing, then subtracting 
each member from unity, again reducing, and finally divid- 
ing one result by the other, we obtain 

sin {Li — r) + sin (/i — r) _ tan Ag + tan A, 
sin (/a — r) — sin (Zi — v) ~ tan Ag — tan A, * 

or by reduction formulas, page 0, (Book of Formulas) 



(18) 



(10) 



(20) 



tan 



L+h 







— V 



tan i ih- h) 



sin (Ag + AJ 



(21) 



sin [A 2 — Ai) ' 
is found from either of equa- 



from which v can be found; / 
tions (18), when v is known. 

To find L, m, e, andp, we proceed as in the determination 
of the table of epochs in the case of the sun, using a similar 
equation, thus: 

L + m fi = i\ — 2e sin (Vi ~p), 
L + ni U = v-i — 2 ^ sin (rg —p), 
L + in ts = i?3 — 2e sin {v^ —p), 
L + m ti — Vi^ — 2e sin {v^ — p) ; 



(22) 



in which 



Vi = V -{- tan 



_j tan (h — y) 



cos I 
and similar values for V2, v^, and v^. 



(23) 



14 PRACTICAL ASTRONOMY. 

To find the ecliptic longitude of perigee V O, represented 
by Pi. we have from the right-angled triangle NPO, 

tan NO — tan {p — r) . cos /, (24) 

from which, 

Pi^r -}- tan-^ (tan (p — r) . cos /). (25) 

Similarly the mean ecliptic longitude of the moon at the 
epoch Xi, is 

Li = r + tan~^ (tan {L — r) . cos /). (2(;) 

To find the mean motion m, we have 

in which s is the length of the sidereal period in mean solar 
days. 

The fxidiiis vector r, is found from 

GO 

and a the mean distance from 

1 + ecos (v —p) 

2. The Ephemeris of the Moon. — The motion of the moon 
is much more irregular and complicated than the apparent 
motion of the sun, owing mainly to the disturbing action of 
this latter body. But this and other perturbations have 
been computed and tabulated, and from these tables, includ- 
ing those of the node and inclination, the places of the moon 
in her orbit are found in the same way as those of the sun 
in the ecliptic. The mean orbit longitude of the moon and 
of her perigee are first found and corrected; their difference 
gives her mean anomaly, opposite to which in the appropri- 
ate table is found the equation of the center, and this being 
applied with its proper sign to the mean orbit longitude gives 
the true orbit longitude. 

The Right Ascension and Declination of the Moon can 
now be computed for any instant of time, thus: subtract the 
longitude of the node from the orbit longitude of the moon, 
and we have the moon's angular distance from her node. 



PRACTICAL ASTRONOMY. 15 

represented in the figure by XM^. This, with the inclination 
/. will give us the moon's latitude and the angular distance 
A'il/i; the latter added to the longitude of the node will give 
the moon's longitude VM^. The latitude, longitude, and ob- 
liquity of the ecliptic suffice to compute the right ascension 
and declination. The radius vector, equatorial horizontal 
l>arallax. apparent diameter, etc.. are computed as in the 
case of the sun. 

THE EPHEMERIS OF A PLANET. 

From the tables of a planet its true orbit longitude as 
seen from the sun is found, as in the case of the moon as 
seen from the earth. The heliocentric longitude and latitude, 
and the radius vector are found from the heliocentric orbit 
longitude, heliocentric longitude of the node, and inclination, 
in the same way as the geocentric elements of the moon are 
found from similar data in the lunar orbit. 

To pass from heliocentric to geocentric coordinates, let P. 
Fig. 2, be the planet's center. E that of the earth. S that of 
the sun, and O the projection of P on the plane of the 
ecliptic. S^^ and EV are drawn to the vernal equinox; 
then let 

r = ES, be the earth's radius vector: 

/•' = SP, be the planet's radius vector; 

A = VS O, be the heliocentric longitude of planet : 

A' = VEO, be the geocentric longitude of planet: 

^ = PS O, be the heliocentric latitude of planet : 

H' =PEO, be the geocentric latitude of planet: 

S = OSE, be the commutation; 

O — SOE, be the heliocentric parallax; 

E = SEO, be the elongation; 

L = VES, be the longitude of the sun; 

/•"= EP, be the distance of planet from the earth. 
To find the geocentric longitude, 
SO=r' cos 6. 
VST=VES = 3m°-L, 

S=1S0°- (360° ~L)~X^L~ 180° - A, (30) 

from which S is known. 



16 PRACTICAL ASTRONOMY. 

In the plane triangle OES, we have 

/•' cos t) + r : r' cos d - r :: tan i(E-h O) : tan i (E - O), (31) 
. S+ + E=mf, {:]2) 

i(E+O) = 90°-^^, (:3:|) 

hence 

tan i{E-0)= cot ^ ;S C^'*^^ ^ ~ '' . (34) 

" . r' cos 6 -\- r ^ ' 

and placmg 

r' cos 
tanj9 = r"^' (''''^) 

we have 

tan i {E - ()) =Q0ii8 J^I^..^ — | = cot 4 6^ tan ( p - 45°) (3c. ) 
^ ^ ^ tanp + 1 - ^^ ^ V 

therefore E and O are known: and we have, 

r = E- (360° -L)=E + L- 360°. (37) 

To find the geocentric latitude, we have 

PO ^^£"0 tan 6*' = >S0 tan ^ (3S) 

tan 6*' ^ ^ ^ sin^ 
tan'^ ~ E0~ sin >S' * 
whence 

tan 6' = tern ^^4^^. (40) 

sm >S ' 

To find >•", we have 

EO=:r'' cos 6% 

SO=^r' cos d. 

In the triangle ES O, we have 

r" cos /?' : r' cos 6* :: sin S : sin ^, 
whence 



(3!)) 



,, , COS sin S 
COS 6' sin £7' 



(41) 



With these data we can readily find the right ascension, 
declination, horizontal parallax, and apparent diameter as 
in the case of the sun and moon. 



PRACTICAL ASTRONOMY. 17 

INTERPOLATION. 

Interpolation. — Whenever the differences of the quan- 
tities recorded in the Ephemeris tables are directly pro- 
portional to the differences of the corresponding times, 
simple interpolation will enable us to find the numerical 
value of the quantity in question. When this is not the case, 
the value is determined by the ''method of interpolation by 
differences." Bessel's form of this formula, usually em- 
ployed, is 

F. = F+ud,+ " *";- ^> rf, + " (" - J) (" - *> rf, + 

(n + 1) n (n-1) (n-2) , , ^ ,,^, 

In this formula F,„ is the value of the function to be deter- 
mined; F, the ephemeris value from which we set out; 
<l^. do, dg, etc., are the terms of the successive orders of differ- 
ences, determined as explained below; a is the fractional 
value of the time interval, in terms of the constant interval 
taken as unity corresponding to which the values of the 
function F are computed and recorded in the tables. To 
use this formula, draw a horizontal line below the value of 
F from which we set out, and one above the next consecutive 
value taken from the ephemeris. These lines are to enclose 
the values of the odd differences di, (/», d,,, etc. The values 
of the even differences d^, d^, d^, etc., being each the mean 
of two numbers, one above and one below in their respective 
(•olumns, are then inserted in their proper places. The follow- 
ing example is given to illustrate the application of Bessel's 
formula. 

Find the distance of the moon's center from Regulus at 
ti P. M. West Point mean time March 24th, 1891. 

The longitude of West Point is 1.93 hrs. west of Green- 
wich; hence the Greenwich time corresponding to 9 P. M. 
West Point mean time is lo.9o hrs. Referring to pages 54 
and 55 American Ephemeris we take out the following data, 
namely: 



18 



PRACTICAL ASTRONOMY. 



A 
March 24. 


F 


ch 


do 


„.; 


<i 


6^ 


27° 


01' 24" 














1° 2S' 9" 








9^ 


28° 


29' 33" 


1' 28' 20" 


+ 13" 


-1" 




12^ 


29° 


57' 53" 




+ 12" 
(+11".5) 




<j 


13^93 


30° 


54' 10".48 


1° 28' 32" 


-1" 





15^' 


31° 


26' 25" 


1° 28' 43" 


+ 11" 


-1" 


" 


18*^ 


32° 


55' 8" 


1° 28' 53" 


+ 10" 






21^ 


34° 


24' 1" 






■ 





Whence, substitutmg in the formula, we have 

i^ = 29° 57' 53" + 0.643 (1° 28' 32") + 0.G43 (- o|^) (11".5) 
+ (0.643) i(- 0.357) i (0.143) (-1"). 
==29° 57' 53" + 56' 18".796 - 1".32, 
= 30° 54' 10". 476 the required distance. 

CONSTANTS OF INSTRUMENTS. 



Equatorial Interval and Run of R. A. Micrometer.— The 

mean of the times of transit of a celestial body over the 
several wires of a transit instrument is called the time of 
transit over the mean of the wires or mean wire. The mean 
wire does not usually coincide with the middle wire, due to 
the impossibility of grouping the wires in perfect symmetry 
with reference to the middle. Since the collimation error 
is usually measured from the middle wire, it becomes 
necessary to determine the distance in time of the mean 
from the middle wire. This is obtained from the '"Equa- 
torial Intervals.'' 

By the equatorial interval of a given wire is meant the 
interval of sidereal time in seconds required for a star on 
the celestial equator to pass from this wire to the middle 



PRACTICAL ASTRONOMY. 19 

wire, or vice-versa. If the interval for a star on the equator 
is known, that for a star of any other declination is found 
hy dividing by the cosine of the declination; these intervals 
may also be used to reduce observations taken on a side wire 
to the middle wire and to supply times of transit over certain 
wires when actual observation has been prevented by clouds 
or other cause. We must therefore determine for each in- 
strument the Equatorial Intervals. 

Equatorial Intervals. — 1st method. Note the actual time 
required for a star of known declination to pass from the 
wire in question to the middle wire and multiply by the 
cosine of the declination: the mean of many such obser- 
vations will be the interval sought. 

M method. By the R. A. Micrometer. The micrometer 
head carries a movable wire which is always parallel to the 
fixed wires of the transit and, as it moves across the field of 
view, it coincides with each of them in succession. The value 
in time of one revolution of the micrometer head can then 
be readily determined and thus that of the interval desired. 
The mean of many determinations is the value adopted. 

The principles involved in both methods are essentially 
tlie same and it will suffice to explain in detail the process 
of finding the value of one division of the micrometer head 
in the case of a portable field transit. 

First, we should be sure that the middle wire (and there- 
fore every one) is truly vertical w^hen the axis is level. This 
may be ascertained by causing an equatorial star to pass 
through the field and noting whether it will thread the hori- 
z(mtal wire during the whole of its passage; if not, move the 
box carrying the wire-frame by the proper adjusting screws 
until the adjustment is perfect. 

Then, shortly before the time of culmination of some slow- 
moving, (circumpolar) star, set the instrument so that the 
star will pass through the field. Set the micrometer wire at 
some even division and on that side of the field of view 
where the star is about to enter. Note the reading of the 
micrometer head and record the time of transit of the star 
over this thread, using a sidereal chronometer whose rate is 
well determined. Set the wire again a short distance ahead 



20 PRACTICAL ASTRONOMY. 

of the star, note the reading and record the time of transit. 
We may thus "step" the micrometer screw throughout the 
whole of its length. 

Then if S be the apparent declination of the star for the 
date, / the sidereal interval between any two observations, 
(corrected for rate if appreciable), R the value of a revolution 
(or division) of the micrometer and M the number of revolu- 
tions (or divisions) corresponding to I, we will have 

sin / = sin / cos S, and R= —- ; (43) 

/" denoting the equatorial value of I in time. If S be less 
than 80° the "correction for curvature of path," involved in 
the trigonometric functions of the intervals, need not be 
applied, so that the equation w^ill read 

/cos_^ 

No correction for difference of refraction between the two 
positions of the star is required, since at its meridian passage, 
the star is moving almost wholly in azimuth. 

By examining these equations we see why a circumpolar 
is preferable to an equatorial star, since if a slight error be 
made in determining I, it will have comparatively little 
effect on i, for cos d will be very small. Moreover we are 
less liable to make errors with a slow moving star, since we 
can estimate with great accuracy the instant of its bisection. 
For handling the very small arcs of Eq. (4:3), "Shortledge's 
Logarithmic Tables" are almost indispensable. 

Since our observations have given us a series of consecutive 
short intervals, we may either use them as such and thus 
determine possible differences in the value of a revolution 
(or division) for different points of the screw, or we may 
combine several short intervals into one long one and thus 
determine the average value for any desired length of the 
screw. The final result should be the mean of several deter- 
minations. 

It is to be noted that in multiplying sin I (or I) in Eqs. 
(43 and 44), by cos ^ we reduce that interval to the equator; 
that is, the product is the time that the star would have 



PRACTICAL ASTRONOMY. 21 

occupied in passing from one position of the wire to the other 
had its declination been instead of S. 

Having thus found R, the Equatorial Interval of any wire 
is found by noting the number of revolutions required to run 
the micrometer wire from the wire in question to the middle 
wire, (u^ing again the mean of several runs.) and multiply- 
ing this by the value of a revolution. 

In using the first method it is to be remembered that the 
observed intervals are to be multiplied by cos S. Conversely 
the actual intervals for any star is found from 

• 7^ sin / . . „ .^^. 

sin /= , =sm t sec o. (4o) 

cos d ^ ' 

or 

I— ,. = / sec S. (46) 

cos 6 ^ ' 

The best stars to use are a, 6. and (3, Ursae Minoris: their 
declinations being accurately given in the Ephemeris, the 
first two for every day and the last one for every ten days. 
The first two require the ''Correction for Curvature." the 
last does not. It a or 6 be used, each observation should 
be compared with one near the middle of the field in getting 
the intervals. Hence for all reasons y^ is the best star to use. 

Reduction to the Middle Wire. — Having now the Equa- 
torial Intervals, we may deternnne the distance in time 
from the mean to the middle wire; or as it is called, "the 
Reduction to the Middle Wire." Suppose the instrument 
to have seven wires, and to be in good adjustment. A star 
at its upper culmination will apparently move over these 
wires from west to east: therefore (with the instrument in 
a given position, say with ''illumination east") let the wires 
be successively numbered from the west towards the east. 

Let a star whose declination is 6 pass through the field, 
and let fi, to, t^, t^, t^, f^, t-,, be the accurate instants of pass- 
ing the corresponding wires; let /i, i^, is, h, ib, U, h, be the 
equatorial intervals from the middle wire. Then the time 
of passing the mean wire is 

^1 + ^2 + ^3 + ^4 + ^5 + ^6 + ^7 



22 PRACTICAL ASTRONOMY. 

The time of passing the middle wire is either 

ti + 7*1 sec S, ti + 4 sec S, t^ + i^ sec ^, t^, 4 — 4 sec S, 
U — «6 sec (J, or f; — ^^ sec ^, 

(note the minus sign in the last three.) Hence the most 
probable time of passing the middle wire is 

2(t-^isecS) :Et ^i ^ ^ , 

— ^ — ^ -= -y- + - Y ^^^ • ^^^^ 

The difference between this and the time of passing the 
mean wire is evidently the second term, or 

^^ sec 6 ^ (^^ + ^^ + ^3) - (4 f ^0 + /.) ^^^ ^ (^^^ 



7 7 

The equatorial value of this reduction, (one of the con- 
stants of the instrument) will then be 

^?: 

and for any given star, the actual interval will be this value 
multiplied by sec ^. Hence we have the rule: To the mean 
of the times add /^i sec (J, noting the signs of both factors. 
The sign of A i is changed by reversing the axis of the in- 
strument. 

Value of one Division of the Level. — The best method 
of determining this quantity in case of a detached level 
is by use of the "Level-trier," which consists simply of 
a metal bar resting at one end on two firm supports, 
and at the other on a vertical screw. Then if d be the 
distance from the screw to the middle of the line joining the 
two fixed supports, and b the distance between two threads 
of the screw, (obtained by counting the number of threads to 
the inch,) the inclination of the bar to the horizon would be 

changed by ^r—- — tt, due to one revolution of the screw. The 

level is then placed on the bar and the number u of divisions 
passed over by the bubble due to one turn (or division) of 
the screw is noted. The value of one division of the level in 

anffle is then — j~. — -r/ . The mean of several observations. 



PRACTICAL ASTRONOMY. 23 

using both ends of the bubble, should be adopted. The value 

in time is ^ , ^ -,, . If no level-trier is available, the level 
15 /id sml 

should be placed on the body of the telescope connected with 
a vertical circle reading to seconds: as for example the me- 
ridian circle of a fixed observatory. Move the instrument 
slowly by the tangent screw and note the number of level 
divisions corresponding to a change of 1" in the reading of 
the circle taking the means as before. By either method the 
level may be tested throughout its entire length. 



The Attached Level and Declination micrometer of the 
Zenith Telescope. — Since in this case the level is attached to 
a circle which does not read to seconds, neither of the pre- 
ceding methods is applicable to the zenith telescope. Again, 
since the declination is at right angles to the R. A. microm- 
eter wire, stars at their meridian transit will not cross it. 
The usual method in this case is to find a value of a division 
of the level in terms of a revolution of the micrometer head. 
Then after finding the latter (which involves the former) we 
may find the actual value of a division of the level in seconds 
of arc. 

Direct the telescope to a small, well-defined, distant, ter- 
restrial object, and set the level so that the two ends of the 
bubble will give different readings. Bisect the object witli 
the micrometer wire, note the reading, also that of each end 
of the bubble. Move the telescope and level together by the 
tangent screw until the bubble plays near the other end of 
the tube. Again bisect the mark by the micrometer wire 
and note all three readings as before. The mean of the 
number of divisions passed over by the two ends of the 
bubble is then the number of divisions passed over by the 
bubble. The difference of the micrometer readings is the 
run of the micrometer. Dividing the second by the first, we 
have the value of a division of the level in terms of a revo- 
lution of the micrometer. Take a mean of several determi- 
nations and denote it by d. 

We can now find the value of one division of the microm- 
eter. For reasons stated when treating of the R. A. microm- 



24 PRACTICAL ASTRONOMY. 

eter, we use a circumpolar star, and at the instant that its 
path is perpendicular to the wire in question. This requires 
us to take the star at its elongation. Manifestly the same 
principles apply to the two cases, since the principal differ- 
ence is that the star and wire have each been apparently 
shifted 90°; the motion of the star with reference to the wire 
not having changed. Some changes in detail are however 
necessary. In the first place, since the motion of the star is 
almost wholly in altitude, we cannot as before neglect differ- 
ences in refraction between two transits. Again, since the 
pressure of the hand in working the micrometer head is in 
a direction to cause a possible disturbance of the instrument 
even though firmly clamped, we must read the level at every 
transit, and if any change has occurred, correct the microm- 
eter readings accordingly. 

As a preliminary, we must determine the time of elonga- 
tion, (in order to know when to begin our observations,) and 
the setting of the instrument, i. e., the azimuth and zenith 
distance of the star at the time of elongation. The hour 
angle is found from 

cos 4 = cot S tan ^, (49) 

from which the sidereal time of elongation is given by 

T, = a±t,-l^T, (50) 

in which a is the star's apparent E. A. for the instant, and 

A T is the error of the chronometer. The plus sign is used 

for western and the minus for eastern elongations. 

The azimuth is given by 

A COS S ^^^. 

sm A = , (51) 

cos cp 

and the zenith distance by 

sin (p ,^„. 

cos Zq — -^-~\ . (52) 

sm d ^ ' 

Set the instrument in accordance with these coordinates 
20 or 30 minutes before the time of elongation, and as soon 
as the star enters the field, shift the telescope if necessary 
so that it will pass nearly through the center. 

The observations are now conducted in exactly the same 
manner as for the R. A. micrometer, with the addition that 



PRACTICAL ASTRONOMY. 25 

each end of the level bubble is read in connection with each 
transit. Then, as before, we take any two sets of observa- 
tions, and compute the value of a revolution, remembering 
that the result must be in arc, and not in time, and that if 
the level is different at the two observations, (31) the number 
of revolutions of the micrometer must be corrected ac-cord- 
ingly. For instance, if the level shows that between the two 
observations the telescope had moved with the star in its 
diurnal path, then evidently the micrometer will indicate 
only a part of the angular distance between the two positions 
of the star, and the level correction must be added to the 
micrometer interval. Conversely, if the telescope has moved 
aga inst the motion of the star. This level correction is found 
as follows: if d is the value of one division of the level in 
terms of a revolution of the micrometer and L the number 
of divisions which the level has shifted, then Ld will be 
the value (in micrometer revolutions) of the correction to be 
applied to M. The method of finding d has already been 
explained. 

It should be noted that if the star is within 10° of the pole, 
each observation must be compared with the one made 
nearest to the time of elongation, since the formula 

sin i — sin I cos S, 
is strictly true only when the wire, at one of the times, co- 
incides with a circle of declination; i. e., when it is at the 
point of greatest elongation. 

The value of a revolution [E), as thus found, is to be cor- 
rected for differential refraction by subtracting from it R A r, 
in which A r is the change of refraction at zenith distance 
Zo (that of elongation) for 1' of zenith distance, and R is 
expressed in minutes of arc. 

The complete formula for the value of one revolution of 
the declination micrometer is then 

„ sin J cos (J -^ ,^„. 

^'= M±Ld --^^'•' (^^) 

in which I is expressed in seconds of arc, M in revolutions, 
L in divisions of the level, R^ in seconds of arc, R is the 
first term of the second member reduced to minutes of arc 



26 PRACTICAL ASTRONOMY. 

and A 7^ is taken from tables. The actual value of a division 
of the level in seconds of arc is then evidently R^ d. 

The preceding method of finding these two constants of 
the zenith telescope is regarded as the best; but provision is 
made in the construction of the instrument for turning the 
box containing the wire frame through an angle of 90°. When 
this is done, the declination micrometer becomes virtually a 
K. A. micrometer, and the value of a revolution may be 
found as described for that micrometer, and then the box 
revolved back to its proper place and clamped. In this case 
however the result must be in arc. The level constant must 
be found as just described. 

REFRACTION TABLES. 

A ray of light passing from a celestial body to a point on 
the earth's surface, may be supposed to pass through suc- 
cessive spherical strata of the atmosphere, the densities of 
which continually increase toward the center. Hence, at 
every surface of augmented density, it is refracted toward 
the normal, and makes the apparent greater than the true 
altitude. 

We have from optics, 

sin qj 



sm q) 

and since in this case fx is positive and greater than unity, 
qt will be larger than cp'. Now the sine of an angle varies 
less rapidly for equal changes in the angle, as the latter 
approaches 90°. Hence it is seen that (p must always receive 
a greater increment to produce a given change in its sine 
than must cp' to produce an equal change in its sine; and this 
divergence grows more marked as q) approaches 90°. That 
is, as the angle of incidence increases or the altitude 
decreases, the refraction increases. 

Again, the difference between q) and q)' will be greater 
as }A. increases; but /i increases with the density of the 
refracting medium. Hence we have the three facts: 
1st, refraction increases the apparent altitude of an object; 



PRACTICAL ASTRONOMY. 27 

2d, it increases in amount with the zenith distance; and 
3d, with the density of the air, which latter depends upon 
its pressure and temperature. 
The adopted value of the refractive function is 

r = a /3^ y^ tan Z, 

In this equation r is the refraction; A and A are quanti- 
ties varying slowly with the zenith distance; /? is a factor 
depending on the pressure, and y upon the temperature of 
the air; Z is the apparent zenith distance; fi therefore 
depends upon the reading of the barometer, and y upon the 
reading of the thermometer. But since the actual height 
indicated by a barometer depends not only upon the pressure 
of the air, but upon the temperature of the mercury, ^ is 
really composed of two factors B and T, the first of which 
depends upon the actual reading of the barometer, and T 
involves the correction due to the temperature of the mercury. 

Nearly all the collections of astronomical tables contain 
''Tables of Refraction," from which may be found the 
various quantities in the equation 

r = a{B T)^ Y^ tan Z. 

The first portion of the table consists of three columns giving 
the values of A, A, and log a, with the apparent zenith 
distance (Z) as the argument. 

The second part contains B, with the height of the barom- 
eter as the argument. The third part gives the value of T 
with the reading of the attached thermometer as the argu- 
ment, and the fourth part gives y with the reading of the 
external thermometer as the argument. Z is the observed 
zenith distance. A substitution of these quantities gives the 
refraction, which must then be added to Z to give the true 
zenith distance. 

The attached thermometer gives the temperature of the 
mercury of the barometer. The external thermometer should 
be screened from the direct and reflected heat of the sun, but 
be so fully exposed as to give accurately the temperature of 
the external air. 

A similar table is sometimes given for passing from true 
to apparent zenith distances. The mode of using is exactly 



28 PRACTICAL ASTRONOMY. 

the same, subtracting the resulting refraction from the true 
zenith distance to obtain Z. It is of use in "setting'' in- 
struments for observation. 

^A ''Table of Mean Refractions'' is also given in nearly 
every collection, and contains the refractions for a temper- 
ature of 50° F., and 30 in. height of barometer, with apparent 
zenith distances or altitudes, as the argument, which may 
be used when a very precise result is not required. 

The above relates only to refraction in altitude. But a 
change in a star's place due to refraction will in the general 
case cause a change in its observed R. A. and Dec. In 
order to ascertain these, two coordinates as affected by 
refraction at a given sidereal time T, we first compute the 
body's hour angle from f— T— R. A., and then its true zenith 
distance (Z) and parallactic angle (^) from the astronomical 
triangle, knowing t, qj, and d. Then if r denote the refrac- 
tion in altitude, found as just explained, the refraction in 
declination will be 

/\d — r cos ^, 
and the refraction in R. A., 

r sin ^ 



A<a: = 



COS 6 



TIME. 

The perfect uniformity with which the earth rotates on its 
axis makes its motion a standard regulator for all time pieces. 
No clock or chronometer can run with perfect uniformity 
and therefore the time indicated by them must ever be in 
error. To find these errors at any instant is the object of 
the time problems in Practical Astronomy. 

Time is measured by the hour angle of some point or ce- 
lestial body. If the point be the true Vernal Equinox its hour 
angle is true sidereal time. 

If the point be the mean Equinox, it is mean sidereal time; 
but since the greatest difference between true and mean 
sidereal time can never exceed 2.3 seconds in 19 years, astro- 
nomical clocks are run on true sidereal time. To pass from 



PRACTICAL ASTRONOMY. 29 

true to mean sidereal time, apply the correction known as 
the Equation of Equinoxes in Right Ascension. 

If the point be the Mean Sun its hour angle is mean solar 
time; all solar time pieces are run on mean solar time. 

If the point be the center of the True Sun, its hour angle 
is true or apparent solar time; to pass from true to mean 
solar time apply the correction known as the Equation of 
Time. 

1. To find the Error of a Sidereal Time -piece "by the Sleridian 
Transit of a Star. (See Form 1,) 

The general statement of the problem is briefly this: since 
the time-piece, if correct, ought to indicate the R. A. of the 
star at the instant of culmination, the difference in time is 
the error required. For the practical solution however it is 
necessary to find by observation the quantities in the follow- 
ing equation and solve it; the transit instrument is supposed 
to be approximately in the meridian. 

a=T+E+aA-^bB-\-cC. (54) 

in which 

a is the apparent R. A. of the star at the date. 

T is the clock time of transit on middle wire deduced 
from the mean of the times on all the wires, plus the 
correction necessary to pass from the mean to the 
middle wire, due to lack of symmetry in position of 
wires. 

E is the clock error, + when slow, — when fast. 

a is the azimuth error of instrument, in time 

/; is the level error of instrument, in time. 

c is the collimation error of instrument, in time (requiring 
a correction if diurnal aberration be considered). 

A is equal to ^^ — ? ^ being the latitude, S the dec- 
lination. 

D • IX COS (cp — S) 

B IS equal to ^-^^ — - . 

^ cos o 

C is equal to sec 6. 



30 PEACTICAL ASTRONOMY. 

We have then to determine by observation and computa- 
tion all the quantities in the second member of Eq. (54) save 
E, Their sum subtracted from the Ephemeris value of a 
will give E. It is well to observe the following order and 
directions: 

1. Enter the number, kind and maker of the chronometer, 

2. Similarly for the transit used. 

3. State whether the illumination be East or West. 

4. Give the star's name from the Ephemeris. 

5. Take the reading of each end of the bubble of the strid- 

ing level, placed on the transit axis. Reverse the 
level and read again. Record under the proper letters 
E. and W. One-fourth the difference between the sum 
of the west and the sum of the east readings is the true 
level reading, the greater sum corresponding to the 
higher end of the axis. This multiplied by the value 
of one division of the level gives h, positive if west end 
is higher. 

6. Record to quarter seconds the instant of the stars 

bisection by each wire. 

7. The reduction to middle wire is the sum of the equatorial 

intervals of the wires, divided by the number of wires, 
multiplied by sec d, observing that the intervals of the 
east wires are essentially negative. This is to be added 
(algebraically) to the mean, and the result T, is the 
chronometer time of passing the middle wire. But due 
to instrumental errors, the middle wire may not be in 
the meridian exactly. In the first place one end of the 
axis may be higher than the other, giving a level error 
6, and the level correction hB. Again, the middle wire 
may not be exactly in the line drawn from the optical 
center of the object glass perpendicular to the axis, 
giving a collimation error c and a collimation correction 
c C. Again the axis may not be truly east and west, 
giving an azimuth error a, and an azimuth correction 
aA. 
The correction bB has already been explained; c may 
be found as in the case of the surveyor's transit, 
using a point so far distant that its image at the stellar 



PRACTICAL ASTRONOMY. 31 

focus will be well defined, and measuring the actual 
error with the micrometer. The transit should be 
accurately horizontal. 
s. A better way is to direct the transit upon some circum- 
polar star, and observe its transits over the wires on 
one side of the center. Reverse the instrument, and 
observe the transits over the same wires again. If T 
and T' are the mean of the chronometer times of transit 
over the middle wire, deduced, by means of the equa- 
torial intervals, from the several observations in the 
first and second positions of the instrument respectively, 
b and 6' the level errors in the two positions, then for 
the first position we have 

c = ^{T'-T)-\-^{h'-h) cos {cp-d)-^ (55) 

€ is thus positive when the middle wire is west of its 
proper position, and its sign is changed by interchang- 
ing pivots. For strict correctness, c should receive a 
correction due to diurnal aberration of — • 021^ cos q)\ cC, 
thus becomes known. 
!). For the determination of a, we must use two of the list 
of stars by which we are determining the chronometer 
error. If both are observed at the upper transit, they 
should differ as little in R. A., and as much in Decli- 
nation as possible. If one be sub-polar, they should 
differ by as nearly 12 hours in R. A., and as little in 
Declination as possible. Then having determined T 
for each star, add to it the level and collimation cor- 
rections. Call the results t and t^; a and a' being the 
apparent R. A. and d and d' the apparent declinations, 
we have 

cos (?>(tanc^-tan(^')' ^^ 

and a A thus becomes known. 

10. Adding these three corrections, hB, cC, and a A to T 

and subtracting the result from a, we have E, the error 
of the chronometer. 

11. For very accurate work, such as is required in connection 

with the telegraphic determination of longitude, it is 



32 PRACTICAL ASTRONOMY. 

usual to employ about ten stars for each determination 
of time, half of the stars being observed with the in- 
strument reversed; and of each half, two should be 
circumpolar, and three equatorial stars. The obser- 
vations are then reduced by the method of least squares ; 
for the mode of making the reductions see pp. 228 and 
229, Professional Papers Corps of Engineers, No. 12, and 
for extended examples, see Report upon the Determi- 
nation of the Astronomical Coordinates of Cheyenne, 
Wyoming, and Colorado Springs, Colorado, in 1872-73, 
"Wheeler Survey." 
12. If only a single star is used, it should be one given in the 
Ephemeris, and which passes near the zenith; since at 
the zenith a A disappears, and this is the only one of 
the three corrections which requires star observations 
for its determination. 

2. To find the Error of a Mean Solar Time -piece by a 
Meridian Transit of the Sun. (See Form 2.) 

Apparent ISToon at any place is the instant of culmination 
of the sun's center at that place. This epoch is expressed in 
three different times, viz. : 

Apparent time, or 12 o'clock apparent time. 
Mean time, or 12 o'clock plus the equation of time. 
Clock time, or that indicated by the mean solar time-piece. 
Note the order and directions that follow: 

1. The mean of all the observed times is the chronometer 

time of transit of sun's center over the mean of the wires. 

2. The reduction to middle wire, as well as the three correc- 

tions, are found as in Form 1. By adding them to the 
above mentioned mean, we have the chronometer time 
of apparent noon. The declination of the sun, used in 
computing these corrections, is to be taken from the 
Ephemeris, allowance being made for the observer's 
longitude. Use Page I Monthly Calendar. 

3. The mean time of apparent noon is 12 hours ± e. In 

computing £ use Page I Monthly Calendar, and make 
allowance for observer's longitude. The Ephemeris gives 
the sign of e. 



PRACTICAL ASTRONOMY. 33 

4, Subtract the chronometer tirae of apparent noon from the 
mean time of apparent noon, and the remainder is the 
error of the chronometer: — plus if slow, minus if fast. 

0. Time-pieces at West Point are run on 75th Meridian mean 

time, i. e., 4"" 9^38 slower than local mean time. Hence 
in finding the error at West Point subtract 4"" 9138 from 
12^ ± f, before proceeding with step No. 4. 

3. To find the Error of a Sidereal Time -piece by a Single 
Altitude of a Star. (See Form 3.) 

The solution of this problem consists in finding the value 
of the hour angle ZPS in the astronomical triangle (see 
Fig. 41, page 78, Young,) having given the three sides of the 
triangle, viz. : ZP the complement of the latitude, PS the 
polar distance of the star, and ZS its zenith distance. The 
latitude is supposed to be known, the polar distance is taken 
from the Ephemeris, and the altitude, the complement of the 
zenith distance, is measured by the sextant and artificial 
horizon. The chronometer time at the instant the star has 
the measured altitude is also taken. 

The data substituted in the formula 

• 1 D , /cos ?/i sin (?>i — a) ,^^. 

smiP=4/ ^ — -. — ^, (57) 

\ cos (p sm a ^ ' 

will give the value of P the star's hour angle, which being 
multiplied by 15 will reduce it to sidereal time. Attention 
is directed to the following notes. 

1 . The arc of the sextant being graduated to 10 minutes, it 

will be most convenient to record the times correspond- 
ing to successive changes of 10' in the star's double alti- 
tude. The altitudes will thus be equidistant, and involve 
no measurement of seconds. 
'Z. Any error in the assumed latitude, north polar distance, 
or the measured altitude, will have its minimum effect 
when the star is on the Prime Vertical. The polar 
distance should also be great; hence, if possible, select 
stars whose declinations are less than the latitude and 



34 PRACTICAL ASTRONOMY. 

observe them near the prime vertical, but not at a very 
low altitude. 

3. It is usual to regard a mean of the times as corresponding 

to a mean of the altitudes, which implies that the star's 
motion in altitude is uniform, which, in general, is not 
true. 

The refraction also varies ununiformly with the alti- 
tude. Both the mean of the times, and the refraction, 
therefore require a correction to reduce them to the mean 
of the altitudes, if very accurate work is required. For 
these corrections, see Chauvenet, Vol. 1, pp. 21G-17. 

4. If very accurate results are not required, we may use mean 

refractions, instead of actual refractions which depend 
on the indications of the thermometer and barometer. 

4. To find the Error of a Mean Solar Time -piece by a 
Single Altitude of the Sun's Limb. (See Form 4.) 

This problem does not differ in principle from that of Form 3. 
The center of the sun takes the place of the star in the 
astronomical triangle, and the other modifications are those 
which follow: 

1. The sun should be observed as near to the prime vertical 

as is consistent with obtaining sufficient altitude to avoid 
irregular refraction. 

2. The semi-diameter requires no correction for augmen- 

tation due to altitude. It is given on Page I, Monthly 
Calendar for Greenwich apparent noon, and may be 
assumed to be the same at mean noon. 

3. The sun's declination, (or polar distance) varies quite 

rapidly, and in order to determine this element for the 
instant of observation we must know our longitude and 
the error of our chronometer, to obtain which is the ob- 
ject of the problem. In practice the error will always 
be known with sufficient accuracy to find the change in 
declination between the Greenwich mean noon and the 
time of our observation. This assumed difference in 
hours being multiplied by the hourly change in declina- 
tion gives the correction in declination to be applied to 



PRACTICAL ASTRONOMY. 35 

the Ephemeris declination, which latter should be taken 
from Page II, Monthly Calendar (not Page I). 

4. The same remarks and precautions apply to the correc- 

tions for semi-diameter and the Equation of Time. 

5. The sun's Equatorial Parallax is given on Page 278, 

Ephemeris. Strictly this should be multiplied by the 
ratio p, of the earth's radius at the equator to that at 
the place of observation, to compute the parallax in alti- 
tude. At West Point log p = 9.999368 - 10. 

5. To find the Error of a Sidereal Time -piece by Equal 
Altitudes of a Star. (See Form 5.) 

If the times when a star reaches equal altitudes on oppo- 
site sides of the meridian be noted, the "middle chronometer 
time " will be the time of transit, provided the chronometer 
has run uniformly. Hence we would have 

T + T 
E=a- ^^ " . 

But if the refraction is different at the two observations, 
the ti'ue altitudes will be unequal when the measured are 
equal. Manifestly therefore the middle chronometer time 
requires a correction equal to one half the hour angle due 
to the change in true altitude; — this change being the differ- 
ence of the E. and W. refractions. 

Hence, we have, in full, 



E=a- 



T, + r.. ^ (r, - r^) cos a 



30 cos cp cos 6 sin t 

E being the chronometer error, a the star's K A., T^ and r,v 
the times of observation, r^ and r^. the corresponding refrac- 
tions, and t one half the elapsed time. 

1 . The observations should be as near the Prime Vertical as 
is consistent with obtaining a good altitude. By select- 
ing a star whose declination is nearly equal to the latitude, 
it will be on the Prime Vertical, near the zenith. In this 
case, although the altitudes vary very slowly, we can 
probably avoid the correction for refraction, since the 
east and west observations will come near together. 



36 PRACTICAL ASTRONOMY. 

2. Use an Ephemeris star, set the sextant index at some exact 

division of the scale, and note the times corresponding: 
to successive increments of 10' or 20' in double altitude. 
Use the same altitudes in the next group of observations, 
in the reverse order. 

3. In finding the difference of refractions subtract the west 

refraction from the east, noting the sign. 

4. The correction is to be added algebraically to the middle 

chronometer time; the result will be the chronometer time 
of transit. 

5. If observations are taken in the west first, (to find the 

error at time of lower culmination,) the method is the 
same, taking care to use the apparent R. A. and Decli- 
nation, corresponding to that instant. 

6. To find the Error of a IXEean Solar Time -piece by Equal 
Altitudes of the Sun's Limb. (Eee Form 6.) 

The general principles involved are the same as in the 
preceding method. But since the sun changes in declination 
between the times of the E. and W. observations, equal 
altitudes do not correspond to equal hour-angles. For ex- 
ample, when the sun is moving north, the morning hour- 
angle will be less than the afternoon hour-angle at the same 
altitude. Manifestly therefore the "middle chronometer 
time " requires a correction equal to one-half the hour-angle 
due to the change in declination. Hence we have, including 
correction for unequal refraction. 



2 / 30 cos q) COS 0^ sin t 

+ (^iTtan 9 + 51^ tan ^) 1 . 

The last term in the bracket is called the ' ' Equation of 
Equal Altitudes.^' A and B are taken from Tables with their 
proper signs. K is the hourly change in sun's declination 
at apparent noon, taken from the Ephemeris. 
1. In this method we assume that the chronometer runs uni- 
formly during the interval between the A. M, and P. M. 



PRACTICAL ASTRONOMY. 37 

observations, and that the true sun's motion in R. A. 
and in Declination is also uniform. 

2. The observations are best made when the sun is near the 
Prime Vertical; see Note 2, Form 5, for making the ob- 
servations, substituting in place of the star the sun's 
lower limb. 

;3. The altitude may, if desired, be reduced to sun's center 
by adding semi-diameter, for computing refraction. (See 
Notes 3 and 4, Form 5.) 

4. The sun's declination, hourly change in declination, and 

Equation of Time, all for local apparent noon, are found 
from Page I, Monthly Calendar, by applying the longi- 
tude. K is positive when the sun is moving north. 

5. The logarithms of A and B in the Equation of Equal 

Altitudes, are taken from tables, with the argument 
"Elapsed Time.'' For noon A is always negative; for 
midnight, positive. For noon or midnight B is positive 
when the elapsed time is less than 12 hours. When 
more B is negative, 
f). The correction for refraction and the equation of equal 
altitudes are added to the middle chronometer time, and 
the result is the Chronometer Time of Apparent Noon. 

Time of Sunrise or Sunset.— This problem is precisely 
similar to that of single altitudes, except that the altitude of 
the sun is known and therefore no observation is required. 
The zenith distance of the sun's center at the instant when 
its upper limb is on the horizon is assumed to be 90° 50', 
which is made up of 90°, plus 16' (the mean semi-diameter of 
the sun), plus 34' (the mean refraction at the horizon). The 
resulting hour-angle replaces P in Form 4, 

Duration of Twilight.— The zenith distance in this case is 
108°, as twilight is assumed to begin in the morning or end 
in the evening when the sun's center is 18° below the horizon. 
(See Art. 130, Young.) 

From the solution of the ZPS triangle it can readily be 
shown that the time required for the sun to pass from the 
horizon to a zenith distance z is 



38 PRACTICAL ASTRONOMY. 

^ = Asinyi^^Hl||5^, (58) 

in which ^ and B,' (called the sun's parallactic angles) are 
the angles included between the declination and vertical 
circles through the sun's center for any zenith distance z. 
and for the horizon respectively, and cp is the observers 
latitude. Making z equal to 108° this becomes 



, o • 1, /l-cos 18° cos O^-^O 

from which the duration of twilight for any latitude and any 
season of the year can be found; the values of ^ and ^' are 
given by 

^ sin a> — sin c^ cos 2: ,^.^^^ 

cos ^ — ^-^ , (60) 

cos o sm z : ^ ' 

and 

^/ sin o) ,^ , 

cos^'==: ^. (61) 

cos o ^ ' 

When ^ is equal to ^' then Hs a minimum, and we have, 
after replacing 1 — cos 18° by 2 sin^9°, 

t = j% sin-i(sin 9° sec cp), (62) 

from which the duration of the shortest twilight is found. 
Under the same condition we have from Eqs. (60) and (61). 

sin (^= —tan 9° sin cp; (63) 

from which the sun's declination at the time of shortest 
twilight at any latitude can be found. 

LATITUDE. 

The latitude of a place on the earth's surface is the decli- 
nation of its zenith. The apparent zenith is the point in 
which the plumb line, if produced, at the point of observation 
would pierce the celestial sphere. The central zenith is the 
point in which the radius of the earth, if produced, would 
pierce the celestial sphere. The latitude measured from the 
central zenith is called the geocentric latitude, and that from 



PRACTICAL ASTRONOMY. 39 

the apparent zenith is called the astronomical latitude or 
simply the latitude. The difference between the latitude 
and the geocentric latitude is called the i^eductioii of latitude. 
. The direction of the plumb line is affected by the local 
attraction of mountain masses on the plumb bob, or on 
account of the unequal variations of density of the crust of 
the earth, at or near the locality of the station. The Astro- 
nomical latitude is determined from the actual direction of 
the plumb line, and therefore includes all abnormal devia- 
tions. The Geographical or Geodetic latitude is that which 
would result from considering the earth a perfect spheroid 
of revolution, without the abnormal deviations above re- 
ferred to. 

Form and Dimensions of the Earth.— Before proceeding 
to the latitude problems it is important to derive some neces- 
sary formulas from the form and dimensions of the earth. 
For this purpose, let us assume that the earth is an oblate 
spheroid about ^the polar axis. Let EP'O be a meridian 
section of the earth through the observer's place O; CP' 
the earth's axis; EQ the earth's equator and HH' the ob- 
server's horizon. Let P be the pole of the heavens; Z the 
apparent and Z' the central zenith; q) the latitude and cp' 
the geocentric latitude. The equation of the observer's 
meridian referred to its center and axes is 

a^y'-hb'x' = a'b', (64) 

in which a and b are the equatorial and polar radius of the 
earth. The coordinates of O being x' and y% we have the 
following analytical conditions. 
For the tangent at O, coincident with the horizon, from 

a^yy' -{-b^xx' = a^b^; (65) 

and the normal at O, coincident with the apparent zenith 
OZ, from 

a^y' {x - x') - b^x' (y - y') = 0. (66) 

From Eq. (65), we have 

tan O ^ C = tan (90° - cp) = ^' ; (67) 

a y 



40 PRACTICAL ASTRONOMY. 

whence 

6'x' tan cp^a'y'. (68) 

Substituting in 

a^y'^ + ¥x'' = a'b' (69) 

and eliminating 6 by 

a^ — ¥ 



we have 



, a cos CD 

x' — 



y 



V 1 — e^ sinV 
a (1 — e^) sin cp 



Y (71) 



l/ 1 — e^ sinV J 

Let s be the length of any portion of the meridian; then 
its projection on the major axis x, is 

ds cos OAC — ds^iYL qy—— dx', (72) 

since x' is a decreasing function of the latitude. Differ- 
entiating the first of Eqs. (71), we have 

da.'=_a(lf^!>#^^. (73) 

(1 — e^ smV)^ 

Equating (72) and (73), we have 
and for any other lalitude cp^, 

Let d^ = l°, then dividing (74) by (75), we have 

ds (1 — e^ sinV/)^ 1 — t ^^ sinV/ i /;^..x 

-7- = V, o . o X3 = ^1 — o 2 . 2 nearly, (76) 

ds, (1 — e^ smV)^ 1 — I ^ sm^ 

which, after solving with reference to e^ reduces to 

g2_2 as — ds, ,^^v 

^ ds sin^ — ds^ sin^/ ' 

from which the value of the eccentricity of the meridian 
can be found when the measured lengths ds and ds, of any 



PRACTICAL ASTRONOMY. 41 

two portions of the meridian line, each 1° in latitude, and 
the latitudes (p and ^^ of their middle points are known; 
for the earth, this has been found to be about ((.(KSIGOUT. 

To find the eqnatcyriaJ and polar radii, we have from 
Eq. (74) after making dq) = l"". 

a = p^, (1 - e- sin»3. (78) 

and from the property of the ellipse. 



h = a v\—e\ (79) 

To find the radius of en r rat u re R at any point of the 
meridian. After substituting the values of dx. dy, and d^y. 
taken from Eqs. (71). in the general formula for radius of 
curvature. 

R = ± <^'-'; + <'/"'. (80) 

d.v d-y ^ ' 



we have 



/? = a~ \, f'', ,, : (81) 

(I-c'sinV)^ 



and hence the hnajfh of one degree of latitude at any lati 
tude (p is. 

'In R _ ')7ra \-j^ 

'M\() ~ ':jfi() (1 -7^ sinV)^ 



/^— .>/.n — '.»/-n /I ^^^-v^a^TTs- (^^) 



To find the lencjth of a decp-ee on a seetion perpeiidieular 
to the meridian at any latitude cp we proceed as follows: 
The radius p of the earth at the observer's place, is the minor 
axis, and the equatorial radius a is the major axis of the 
elliptical section, cut out of the earth by a plane perpendicu- 
lar to the meridian plane, passed through the center and the 
observer's place. 

Squaring and adding Eqs. (71) and extracting the square 
root, we have the radius of the earth at the observer's place: 
or (83) 

_ / e^ (1 — e^) sinV _ / 1 — 2 e^ sinV + e* sinV 



42 PRACTICAL ASTRONOMY. 

The square of the eccentricity of the section is 

n— oj^ — P^ _ e^ (1 — e^) sinV . 
a^ 1 — e^ sin^(p 

which being substituted for e^ in Eq. (82) after making 
cp = 90°, gives 

^'=^aA/- ^^^rf^'P. . - (84) 

360 y 1 — e^ (2 — e^) smV 

To find the length of a degree of longitude at any latitude 
(Pf we know, Eqs. (71), that the radius of the parallel is x': 
therefore we have 

2 7r , %n cos ^ 

360 360 y 1 — e^ smV 

The value of the radius of the eaiHh, at any latitude (p, is 
derived from Eq. (83) or, 

/ 1 — ^e^ sin^<z? + e^ sin V 
' \ 1 — e^ siircp 

which, for logarithmic reduction, when a is made unity 
may be placed under the form 

log p = 9.9992747 + 0.0007271 cos 2 ^ - 0.0000018 cos 4 cp. (86) 
From the figure and Eqs. (71), we have 

, , a cos (p ,^^. 

x' = p cos cp' = -7=====- . (87) 

y 1 — e^ siircp 

. . , a (1 — e^) sin cp .^^. 

y' = p sm ^' = \ / . =^ . (88) 

y 1 — e^ smV 

Multiplying these equations by cos ^ and sin ^ respectively, 
adding and reducing we have 



and from (87), 



cos ((p-(p')= Vl -f sinV, (89) 



, a cos <p ..^. 

cos ^ = ^ . =, (90) 



P y\ — e^ sin^^ 
Whence by combination we have 



or 
cos (p' COS {cp — cp')z= —^ COS cp', (91) 



PRACTICAL ASTRONOMY. 43 

and solving with reference to p we have 

p = a4/ "^"T'^f' -^, (92) 

\ COS cp COS (cp— cp')' ^ ^ 

which is capable of logarithmic computation. 

To find the reduction of latihide cp — cp'. Since cp is the 
angle made by the normal with the axis of .r we have 

tan (^ = — ^ , (93) 

and from the figure we have 

tan cp' = ^ . (94) 

Differentiating the equation of the meridian section we 
have 

ij^_ b^ dx 

X a^ dy' ^ ' 



Whence 

a 



tan cp' = -y tan cp= {l — e'^) tan (p. (96) 



Developing into a series, we have 

cp-cp' = ^_-^ sin2(p- [jzrj2j sin 4 (?> + etc. (97) 

But since e=: 0.0816967 this reduces to 

cp — (p' = 690". 65 sin 2 cp — 1".16 sin 4^ very nearly. (98) 

Latitude.— The general problem of latitude consists in 
finding the side ZP in the ZPS triangle, any other three 
parts being supposed given; The following methods are 
those usually employed. 

L By Circumpolars. — This depends on the fact that the 
altitude of the Pole is equal to the latitude of the place. Let 
a and a' be the true altitudes of a circumpolar star at upper 
and lower culmination respectively; p and j^' the corre- 
sponding polar distances, and cp the latitude; then we have 

cp — a—p, cp = a'+p', cp — ^(a + a')+^{p'—p). 



44 PRACTICAL ASTRONOMY. 

The change from p to p' is ordinarily so small in the inter- 
val (12 hours) between the observations as to be negligible: 
it is due solely to precession and nutation. This method is 
free from declination errors but subject to changes and errors 
in the refraction. 

Any instrument by means of which correct altitudes can 
be measured may be employed for this method; as the sex- 
tant, meridian circle, or zenith telescope, if necessary. With 
the sextant the method is applicable, only in high latitudes 
during winter so that both culminations occur during the 
night time. For greater accuracy we must employ circum- 
meridian altitudes and reduce to the meridian as explained 
in the 2d and 3d methods. In selecting a star, one with a 
small polar distance should be preferred to avoid irregular 
refraction, and this need not necessarily be an Ephemeris 
star owing to the small change of polar distance in 12 hours. 
For "Reduction to Meridian " with meridian circle, see Notes 
at the end of 2d method; with sextant, see 2d method. 

2. By Meridian Altitudes or Zenith Distances.— When the 
star is on the meridian the zenith distance z and declination 
6 are the only quantities to be found. If the star culminate 
between the pole and zenith then 

qj — 6 — z\ 

if between the zenith and the equinoctial then 

(p— d + z. 

This method is an exceedingly exact one when the obser- 
vations are made with an instrument, such as the transit 
circle, accurately adjusted to the meridian and the star cross 
very near to the zenith. If the observation be made with a 
sextant it is not possible to select the instant of culmination 
and therefore it is necessary to reduce the observed altitude 
to what it would have been at the instant of culmination. 
Even with a transit circle, it is the rule to take several alti- 
tudes during the passage of the star across the field, and 
reduce them to the meridian, as explained in Notes at end of 
this method. 



PRACTICAL ASTRONOMY. 45 

To Reduce an Altitude, Observed at a given Time, to the 
Meridian. — Substitute 1 —2 sin'^P for cos P in the formula 
derived from the Z PS triangle, 

sin cp sin 6 + cos cp cos S cos P= sin a, 
and it becomes. 

sin (p sin S -^ cos fp cos 6 — 2 cos cp cos S sinH P= sin a. 
or 

cos (fp — 6} = cos z — sin a + 2 cos <^ cos 6 sinHP. 

The above formula is rigorously correct, and applies to a 
body at any distance' from the meridian. 

Since the second member involves q.>, we must either 
know qj approximately, or make our observations quite near 
the meridian. In the latter case, the measured altitudes 
are known as "circum-meridian altitudes." and the reduc- 
tion of such altitudes to the meridian is rendered very sim- 
ple by the special formula. 

, cos Q) cos 6 2 sin'^^P /cos o) cos 6\ ' 2 tan a sin'^P ,, ^, 

a. — a-] , — -. — ~ — ■ — -. — ., (09) 

' cos«^ sm 1 \ cos a^ / sin 1 ^ ^ 

obtained by developing the preceding into a series, and 
neglecting the sixth and higher powers of the verij sniaU 

. , „ ,p, , .-^sin'iP , 2 sin'^^P 

(luantity sm^P. Ihevaluesof . ,/>' and - . — -yj~ are 

^ -^ - sni I" sm 1" 

taken from tables with P as the argument. 

In this formula, a is the true altitude and 6 the declina- 
tion at time of observation; a^ is the desired meridian alti- 
tude and P the hour angle. 

For stars below the pole (e. (/., in method by circumpolars), 
we have simply to reckon P from the time of lower transit, 
and 6 as exceeding 00° by the star's polar distance, which is 
the plan followed in Ephemeris. Cos 6 will thus change its 
sign, and both terms of the reduction will become negative. 

If an approximate value of q) be known (for use in second 
member) that of a^, follows at once from a^ = d + 00° — ^. If 
not, one, may be found as follows: In this method, double 
altitudes are taken in as quick succession as possible from 
a few minutes before, until a few minutes after, meridian 



46 PRACTICAL ASTRONOMY. 

passage. The greatest altitude measured will therefore be 
very near the meridian altitude, and its substitution in the 
preceding formula will give a value of qj sufficiently accu- 
rate for the purpose. 

The mode of making observations and reductions in case 
of a star with sidereal chronometer, will be at once appar- 
ent from an explanation in case of the sun with a mean 
time chronometer. It must be borne in mind that the 
declination of the sun is constantly changing. 

The observations are made as just explained on a limb of 
the sun, viz. : Several double altitudes are taken as near 
together as possible, as many before, as after meridian pass- 
age, and the corresponding chronometer times noted. (Note 
the difference between this, and sextant observations for 
time.) 

Now if we suppose each observation to have been reduced 

to the meridian, after correcting for refraction, parallax 

and semi-diameter, we would have several equations of 

the form 

a^—a-\- Am — Bn, 

2 sin^ ^P 
in which m and n are the tabular values of - — -. — -77-' and 

sm 1 

2 sin* ^P 

— -. — ?77-' and A and B the remainina: factors of the corre- 
sm 1 

sponding terms in Equation (99). Any one of the equations 

will give for the latitude, 

^ = (J + 90° - (a + A m - B n). (100) 

In this equation, S is the declination at the time of obser- 
vation. For, since the reduction to the meridian has been 
made with this value of S in obtaining A and B, -a-^-Am — Bn 
is manifestly the meridian altitude of a body whose dec- 
lination is constantly S. In fact, the reduction to the 
meridian by the formula given, can be computed only on 
the hypothesis of a constant declination. We are thus deal- 
ing with a fictitious sun, whose declination on the meridian 
differs from that of the true sun. But since declination and 
meridian altitude always preserve a constant difference 
(the colatitude), we see that Equation (100) will give the 



PRACTICAL ASTRONOMY. 47 

correct value of <^, due to perfect balance in the errors of d 
and {a-\- A m — B n ) . 

The mean of all the equations due to the several observa- 
tions will be 

qj = S,-i-Uif-{a,-^J,w,-Bono). (101) 

Reasoning in the same manner with reference to the mean 
fictitious sun (with its declination Sq and meridian altitude 
[ao + -^o^^/o— ^o'^o]). we see that the result will be perfectly 
rigorous in theori/ if we employ the mean, S^), of the actual 
declinations, not onh' for the first term in the value of cp, 
but for the single computation of Aq and Bq. We thus avoid 
a separate computation for each observation. 

The result will moreover be perfectly rigorous in practice 
if we use for S^ the declination corresponding to the mean 
of the times; since in the 30 minutes covered by the obser- 
vations, the departure of the sun's declination from a nniforni 
increase or decrease, is negligible. 

We have still to determine the value of P from the chro- 
nometer time of each observation, and in this determination 
it must be borne in mind that P (in arc) is the angular dis- 
tance of the true sun from the meridian at the instant of 
observation. 

There are two reasons why this distance (in time) cannot 
be given directly by a mean time chronometer. First, the 
chronometer will usually be gaining or losing, /. e. , it will 
have a "rate/' Secondly, a mean time chronometer, even 
when running without rate, indicates the angular motion 
of the mean sun, which may be quite different from that 
of the true sun, as shown by the continual change in the 
Equation of Time. 

We therefore proceed as follows: From Page I, Monthly 
Calendar of the Ephemeris (knowing the longitude), take 
out the Equation of Time. Add this algebraically to 12 
hours, apply the error of the chronometer, and the result 
will be the chronometer time of apparent noon. The differ- 
ence between this and the chronometer time of each obser- 
vation, gives the several values of P in time, each subject to 
the two corrections mentioned. To find the correction for 



48 PRACTICAL ASTRONOMY. 

rate, let r represent the number of seconds gained or lost in 
24 hours (a losing rate being positive for the same reason 
that an error slow is positive). Then if P' be the corrected 
hour angle, we will have 

P' : P::864G0 : 86400 -r. [86400==:(50 x (JO x 24]. 
Or 

8 040 
86400-?' • 
Or 

2 sin^ i P' 2 sin'iP / 86400 \^ , 2 sin^ \ P 

n 



sm 1" sm 1" \86400-/7 sm 1" 

Hence we will also have 

J. /?? (corrected for rate) — A; ^ — ^ "4 — |, . 

cos a^ sm 1 ' 

Hence if we compute A by the formula 

. , cos cp cos d 
COS a^ 

we may employ the actual chronometer intervals and pay 
no further attention to the question of rate. 

From k — I -—--tk?. ) ^ values of A' are tabulated with the 

\86400 — 77 

rate as the argument. 

The second correction depends, as just stated, on the differ- 
ence between the motions of the true and mean sun. while 
the former is passing from the point of observation to the 
meridian. In other words it depends on the change in the 
Equation of Time in the same interval, or, which is the 
same thing, upon the rate of an accurate mean solar chro- 
nometer on apparent time.^ 

If therefore we let e represent the change in the Equation 
of Time for 24 hours (positive w^hen the Equation of Time 
is increasing algebraically), it is evident that r — e will be 
the rate of the given chronometer on apparent time, and 
that the corrrection for this total rate may be computed as 
just explained for r, or taken from the same table, using 
r — e as the argument instead of r alone. 



PRACTICAL ASTRONOMY. 49 

The operation of reducing the observations is then, in 
brief, as follows. See Form 7. 

Correct the mean of the double altitudes for eccentricity 
and index error. Correct the resulting mean single altitude 
for refraction, semi-diameter, and parallax in altitude. 
Denote the result by a^. 

From the Equation of Time (Page I Monthly Calendar), 
longitude and chronometer error, find the chronometer time 
of apparent noon. 

Take the difference between this and each chronometer 
time of observation, denote the difference by P. and their 
mean by Pq. 

With each value of P, take from tables the corresponding 
values of m and n . Denote their respective means by //?o and //(,. 

From Page II Monthly Calendar, take the sun's declination 
corresponding to the local apparent time . Po. and denote it 
by (^«. 

If q) can be assumed with considerable accuracy, determ- 
ine the corresponding a^ by a^ ~ (to + 90° — (p. 

If not, take the greatest measured altitude, correct it for 
refraction, etc.. call it a^, and deduce (p from the above 
equation. 

From the rate of the chronometer and change in Equation 
of Time, (both for 24 hours,) take k froin the table. 

With these values of k, cp. a^, and 6^, compute 

. cos (p cos ^0 / 1 T> i i 4. 

Ao = ^ A', and P,, = ^o tan a,. 

cos a, ' 

The latitude then follows from 

(p^6, + 90" — {a, + A, Dio - BoUo). (102) 

With a star observed with a sidereal chronometer, the ob- 
servations are the same, and the reduction is only modified 
by the fact that parallax, semi-diameter, equation of time 
and longitude do not enter, while the declination is con- 
stant. See Form 8. 

If the star lie between the zenith and pole, the formula 
becomes 

(p = (ao + Aoiiio - BoUo) - 90° -F 6,. (103) 



50 PRACTICAL ASTRONOMY. 

1. An Ephemeris star is to be preferred to the sun. since 
the reduction is more simple, its dechnation is better known 
and constant, it presents itself as a point, which is of ad- 
vantage in sextant observations, and we have a greater 
choice both in time and the place of the object to be observed. 
The double altitudes should be taken at as nearly equal inter- 
vals of time and be as symmetrically arranged with refer- 
ence to the meridian, as practicable. 

2. By comparing Eqs. (102) and (103) we see that constant 
errors in the measured altitudes, and in refraction, will be 
nearly eliminated by combining the results of two stars, 
one as much north as the other is south, of the zenith. 

3. By examining the rigorous equation. 

sin a^ — sin a = 2 cos q) cos 6 sin^iP, 

we see that at a given locality and with a given value of P, 
the reduction to the meridian diminishes as we select a star 
with a greater declination. It might therefore seem most 
desirable to select a star near the pole. 

But by rewriting the assumed formula for the reduction, 
expressing the first term as a function of cp and d only, and 
including the third term which has heretofore been omitted, 
we have (Formula 2, P. 4, Book of Formulas), 

_ 1 tan a^ f(l + 3 tan^a^) 

' ~tan^ — tan(J (tan^ — tan^p (tan ^ — tan (J)^ 



_ 2 sin^P _ 2 sinHP ._ 2 sin^ P 

111 — 5 _, //^ , ^1 — i 777 , o — . f-f 

sm 1 sm 1 sm 1 



] 



From this equation it is seen that if a star be selected 
which culminates at a considerable distance from the zenith, 
either north or south, each term of this development is much 
smaller than in case of a star culminating near the zenith, 
either north or south. 

Since the third term has been entirely neglected in the 
previous discussion, it becomes desirable to select our star 
in such a manner that the omitted term (and hence • all 
following it) shall be small; and this, as just seen, will occur 
when there is considerable difference between the latitude 
and the star's declination in either direction. 



PRACTICAL ASTRONOMY. 51 

For example in latitude 40° N., if Ave observe a star at 
declination 0°, the observation may be made at 20'" from 
meridian passage and yet the third term amount only to .01", 
which v7ould affect the resulting latitude by one linear foot. 
Or it may be made at 27"" from culmination, and the third 
term amount only to .1", affecting the resulting latitude by 
ten feet. 

A star at an equal altitude north of the zenith, declination 
S0°, (for combination with the preceding as recommended) 
may be observed at 48 and 02 minutes from culmination, 
with no larger errors. 

With other latitudes the figures will vary, but the principle 
remains the same. 

From an inspection of the third term 

1(1 +3tan-a,) 2 sin ^jP 
(tan q> — tan 6f ' sin \"' 
it is seen that an unfavorable position of the star, causing 
the first factor to be excessive, may be counterbalanced by 
diminishing the hour angle, P. 

Hence the general rule: Select a star whose declination 
differs considerably from the latitude. This Avill give ample 
time for taking a series of altitudes. As the declination of 
the selected star approaches the latitude, restrict the obser- 
vations to a shorter time, greater care in this respect being 
necessary for south stars. 

From the above expression for the .3d term, we may readily 
find the declination of a star, which, with a convenient hour 
angle (say 10 minutes) shall not give an error greater than 
any desired limit, (say .1")- 

4. It must be borne in mind that if the observations are 
made by means of a meridian circle attached to a transit 
instrument, the reduction to the meridian (due both to side 
threads and instrumental errors), will take a different form, 
viz. : 

, . . ^2sinH(P— m) 

+ sm cos o ^.„ — ~ , 

sm 1" 

where m is a function of the level and azimuth errors of the 
transit, defined by the equation, 

m — b cos cp + a sin cp. 



53 PRACTICAL ASTRONOMY. 

The last factor in the reduction may evidently be taken from 
the usual tables, with the argument P— m, instead of P. 

The upper sign applies to stars north of the zenith (ob- 
serving that cos 6 is negative for subpolars), and the lower 
to stars south of the zenith. 

Another form, used in the next method, is 

, . ^ ^ sinH(P— tn) 
sm 1" 

3. Latitude by Opposite and nearly equal Meridian Zenith 
Distances. Talcott's Hethod. See Form 9. 

This method depends upon the principle that the astro- 
nomical latitude of a place is equal to the declination of the 
zenith. 

Let z,, and z^ represent the observed meridian zenith 
distances of two stars, the first north and the second south 
of the zenith; r„ and r, the corresponding refractions; and 
dn and S^ their apparent declinations. Then, q) denoting 
the latitude, 

^ = (^, + 2;,+ /-,, (104) 

From which 

Z Z </ 

Since refraction is a direct function of the zenith distance, 
this equation shows that any constant error in the adopted 
refraction will be nearly or wholly eliminated if we select 
two stars which culminate at very nearly the same zenith 
distance, and provided also that the time between their me- 
ridian transits is so short that the refractive power of the 
atmosphere cannot be changed appreciably in the mean 
time. 

Again, since absolute zenith distances are not required, 
but only their difference, if the stars are so nearly equal in 
altitude that a telescope directed at one, will, upon being 
turned around a vertical axis 180° in azimuth, present the 
other in its field of view, then manifestly the difference of 



PRACTICAL ASTRONOMY. 53 

their zenith distances may be measured directly by the 
declination micrometer, and the use of a graduated circle 
(with its errors of graduation, eccentricity, etc) be entirely 
dispensed with, except for the purpose of a rough finder. 
The instrument used in this connection is called a "Zenith 
Telescope.'' Its construction, and application to the end in 
view, are best learned from an examination of the instru- 
ment itself. 

Again, since errors in the declinations will affect the 
resulting latitude directly, we should be very careful to 
employ only the apparent declinations for the date. 

The following conditions should therefore be fulfilled in 
selecting the stars of a pair: 

1st. They should culminate not more than '^0°. or at most 
25° from the zenith. 

•2d. The}^ should not differ in zenith distance by more 
than 15'. and for very accurate work, by not more than 10'. 
The field of view of the telescope is about 30'. The limit 
assigned prevents observations too near the edge of the 
field, and lessens the effect of an error in the adopted value 
of a turn of the micrometer head. This limit also requires 
a very approximate knowledge of the latitude. 

3d. They should differ in R. A. by not less than one 
minute of time, to allow for reading the level and microm- 
eter, and by not more than fifteen or twenty minutes, to 
avoid changes in either the instrument or the atmosphere. 

Since the Ephemeris stars, whose apparent declinations 
are given with great accuracy for every ten days, are com- 
paratively few in number, it becomes necessary, in order to 
fulfill the above conditions, to resort to the more extended 
star catalogues. 

But since in these works only the stars' mean places are 
given, and those for the epoch of the catalogue (which fact 
involves reduction to apparent places for the date), and 
moreover since these mean places have often been inexactly 
determined, it becomes desirable to rest our determination 
of latitude on the observation of more than one pair. For 
example, on the "Wheeler Survey," west of the 100th 
meridian, the latitude of a primary station was required to 



54 PRACTICAL ASTRONOMY. 

be determined by not, less than 135 separate and distinct pairs 
of stars, these observations being distributed over 5 nights. 

Preliminary Computations.— We should therefore form a list 
of all stars not less than 8th magnitude which culminate not 
more than 25° from the zenith and within the limits of time 
over which we propose to extend our observations, arrange 
them in the order of their R. A., and from this list select our 
pairs in accordance with the above conditions, taking care 
that the time between the pairs is sufficient to permit the 
reading of the level and micrometer, and setting the in- 
strument for the next pair; say at least two minutes. 

A "Programme" must then be prepared for use at the 
instrument, containing the stars arranged in pairs, with 
the designation and magnitude of each for recognition 
when more than one star is in the field; their R. A., to 
know when to make ready for the observation; their 
declinations, from which are computed their approximate 
zenith distances; a statement whether the star is to be 
found north or south of the zenith, and finally the "setting" 
of the instrument for the pair, which is always the mean of 
the two zenith distances. 

The declinations here used, being simply for the purpose 
of so pointing the instrument that the star shall appear in 
the field, may be mean declinations for the beginning of 
the year, which are found with facility as hereafter indi- 
cated. Similarly for the R. A. For this Programme, see 
Form 9. 

Adjustment of Instrument. — The instrument must next be 
prepared for use. The main axis is made vertical by the 
.leveling screws, and the adjustment tested by noting 
whether the striding level placed on the horizontal axis will 
preserve its reading during a revolution of the instrument 
360° in azimuth. The horizontality of the latter axis is tested 
by the level in the usual way. The collimation error and 
verticality of the wires may be determined by stars as in 
case of the transit. The former should be reduced to zero. 
The methods of determining the value of a division of the 
micrometer and level have already been described. 



PRACTICAL ASTRONOMY. 55 

The instrument is placed in the meridian by the aid of a 
sidereal chronometer whose error is known. Point the tele- 
scope on a slow-moving star and bisect it by the vertical 
wire. Retain the bisection by turning the instrument in 
azimuth until the true sidereal time is equal to the star's 
apparent R. A. This operation should be repeated with 
other stars, taking care that the instrument is level. If the 
error of the chronometer be not known, it may be determined 
by observation of a star in the zenith, since no azimuth error 
of the instrument, however large, can affect the time of 
transit of a star in that place. Alternating observations on 
circum-polar and zenith stars will then give the required 
adjustment. When this is perfected, one of the movable 
stops on the horizontal circle is to be moved up against one 
side of the clamp, and there fixed by its own clamp screw. 
The telescope is then turned 180° on its vertical axis, and 
again adjusted to the meridian by a circum-polar star. The 
other stop is then placed against the other side of the clamp, 
and fixed. The instrument can now be turned exactly 180° 
in azimuth, bringing up against the stops when in the 
meridian. 

Observations. — The finder being set to the mean of the zenith 
distances of the two stars of a pair, the bubble of the attached 
level is brought as nearly as possible to the middle of its tube, 
and when the first star of the pair arrives on the middle transit 
wire (the instrument being in the meridian) it is bisected by 
the declination micrometer wire, the sidereal time noted, 
and the micrometer and level read. The telescope is then 
turned 180" in azimuth, the clamp bringing up against its 
stop. The same observations and records are now made for 
the second star. The instrument is then reset for the next 
pair, and so on. The time record is not necessary unless it 
be found that the instrument has departed from the meridi- 
an, or unless observation on the middle wire has been pre- 
vented by clouds, and it becomes desirable to observe on a 
side wire rather than lose the star. In these cases the hour 
angle is necessary to obtain the "reduction to the meridian." 

The observations are recorded on Form 9 a. In the 
column of remarks should be noted any failure to observe 



56 PKACTICAL ASTRONOMY. 

on middle wire, weather, and any circumstance which might 
affect the reliability of the observations. 

Reduction of Observations.— By referring- to Eq. (106) the 
general nature of the reduction will be evident. The principal 
term in the value of q) is d„ + 6^, which as before stated, 
must be found for the date. Since z^ — z,, has been measured 
entirely by the micrometer and level, this term involves 
two corrections to ^„ + ^.s; r„ — r^ involves another, and the 
very exceptional case of observation on a side wire involves 
another. 

1st. The reduction from mean declination of the epoch of 
the catalogue to apparent declination of the date. Let us 
take the case of the B. A. C. (British Association Catalogue.) 

The stars mean place is first brought up to the beginning 
of the current year by the formula 

in which d"=mean north polar distance as given in cata- 
logue, p'== annual precession in N. P. distance, s' = secular 
variation in same, yu' = annual proper motion in N. P. dis- 
tance, (all given in catalogue for each star), 1/= number of 
years from epoch of catalogue to beginning of current year, 
and d'— the mean N. P. distance at the latter instant. To 
this, the corrections for precession, proper motion, nutation, 
and aberration, since the beginning of the year, are applied 
by the formula 

in which refractional part of year already elapsed at date, 
given on pp. 285-293, Ephemeris; A, B, C, D, are the Bes- 
selian Star Numbers, given on pp. 281-284, Ephemeris for 
each day; a' , h\ c', d\ are star constants, whose logarithms 
are given in the catalogue; and c?= star's apparent N. P. 
distance at date. Then (J=90° — d. 

The quantities a', h\ c', d' , are not strictly constant; 
indeed many of their values have changed perceptibly 
since 1850, the epoch of B. A. C. If it be desired to obviate 
this slight error, it may be done by recomputing them by 



PRACTICAL ASTRONOMY. 57 

formulas derived from Physical Astronomy, or, in part, by 
using a later catalogue. In this connection a work pre- 
pared under the '' Wheeler Survey," entitled, ''Catalogue 
of Mean Declinations of 52018 Stars, Jan. 1. 1875," will be 
found most convenient, embracing stars between 10° and 
70^ N. Dec, and therefore applicable to the whole area of 
the U. S. exclusive of Alaska. 

With this catalogue, the reductions are made directly in 
rleclination, not N. P. distance, and by the formulas. 

6 = 6' + TjV -^ Aa' + Bb' + Cc' + Dd\ 

in which everything relates to declination. 
Exactly analogous formulas hold for reduction in R. A. 
•^d. The micrometer and level corrections to 

6s+6„ z,-z, 

"2" : or ^. 

Let us suppose that, with the telescope set at a given 

inclination, the micrometer readings are greater as the body 

viewed is nearer the zenith; and in the first instance, that 

the inclination as shown by the attached level is not changed 

when the instrument is turned 180° in azimuth. 

z . — z 
Then •' " will be given wholly by the micrometer, and 

l)e either Jx, or , E, m which >/?., and m„ are 

the micrometer readings on the south and north stars re- 
spectively, and R the value in arc of a division of the 
micrometer head. Since the readings increase as the zenith 

distance decreases, it is manifest that — "^ — - R is the one of 

the two expressions which will represent ^ " with its 

z 

proper sign. 

But as a rule, the inclination of the optical axis of the 
telescope will change slightly due to the necessary revolu- 
tion between the observations of the stars of a pair, — the 
fact being indicated by a different reading of the level. In 
this case, the difference of micrometer readings will not be 



58 PRACTICAL ASTRONOMY. 

strictly the difference of zenith distance as before, but will 
be that difference + the amount the telescope has moved. 
The micrometer readings therefore require correction be- 

fore they can give - ^' " • Since it is immaterial which 

star of the pair is observed first, let us suppose it to be the 
southern, and let /„ and Z« be the readings of the ends of the 

bubble. Then ^ * will be the reading of the level. Now 

if, on turning to the north, the level shows that the angle 

of elevation of the telescope has increased, the micrometer 

reading on the northern star will be too small, by just the 

amount corresponding to the motion of the telescope in 

altitude; and this whether the star be higher or lower than 

the southern star. Consequently m,^ must be increased to 

compensate. If /'„ and /',, be the reading of the present 

north and south ends of the bubble, then the bubble reading 

/' — /' 
will be -^ — ^ ; the change of level, in level divisions, will be 

In^ + ^-^\^ , and in arc (^" + ^'->)-(^^ + ^'-) /). 

Since, upon turning to the north, the angle of elevation of 
telescope was supposed to increase, this quantity is positive; 
and being the angular change of elevation, it is the correc- 
tion to be applied to in,,. 

If the telescope diminished its elevation on being turned 
to the north, it would be necessary to diminish nin by the 
same amount. But in this case the above correction is 
obviously negative, and the result will be obtained by still 
adding it algebraically. 

m 
The correction to -^ willbehalf the above amount; hence 

in all cases we have the rule. Subtract the sum of the 
south readings from the sum of the north. One-fourth the 
difference multiplied by the value of one division of the 
level, will be the level correction. The true difference of 
observed zenith distances of the two stars, is therefore 

^ nin-m, ^ {ln+l'n)-{ls+l's) ^ 



PRACTICAL ASTRONOMY. 



50 



3d. The correction for refraction, or 



Since the 



stars are at so small and nearly equal zenith distances, 
differences of actual refractions will be practically equal to 
differences of mean refractions (Bar. 30 in., F. 50°). 

dr 



These differences are given by. i\—y„ — {z^—z,) ^, z, and 



z,^ being expressed in minutes and -^ denoting the change 

in mean refraction for a change of one minute in zenith 

distance. It is obtained by differentiating the equation for 

J, ^. ^ 1 , dr a sin 1' 

mean retraction ]- = (x tan z. whence we have -r^ = 7, — , 

dz COS'' z 

a being constant and taken from tables of refraction. The 
following table of values is given, in which we may inter- 
polate at pleasure: 





dr 


z 






dz 


0° 


0.0168" 


5° 


0.0169" 


10° 


0.0173" 


15° 


0.0180" 


20° 


0.0190" 


25° 


0.0205" 



The principal term in z^—z„ is m„ — m^. Hence we have 

— -r " =4 (ni„~mA — - ^— , and the correction for refraction 
2 ^ m dz' 

will have the same sign as the micrometer correction. 

Hence the rule: Multiply one-half the micrometer cor- 

dr 
i-ection ni minutes bv the tabular value of -r- , and add the 

dz 

result algebraically to the other corrections. 

4:th. The correction to the zenith distance for side wire 
observations. 

// the zenith distance of a star between the zenith and the 
equinoctial be measured by an instrument ivhich moves only 
in the meridian, it will have its greatest value when the star 



60 PRACTICAL ASTRONOMY. 

is on the meridian. In any other position of the star, the 

ordinary rule as to relative magnitude, applies. 

But whatever the position of the star, the numerical value 

of the "reduction to the meridian," due to an observation 

on a side wire, is different from that heretofore discussed. 

being in this case i (15 P)^ sin 1" sin 2 d. Hence, if in using 

the zenith telescope, a star be observed south of the zenith 

on a side wire, the above correction must be added to the 

observed to obtain the meridian, zenith distance. Below 

the equinoctial sin 2 <^ will be negative, hence the rule is 

general. North of the zenith, the correction is to be sub- 

z — z 
tracted, as just explained. By inspecting the term ^ " ' we 

see that in any case one-half this reduction, or [0.1347] P' 
sin 2 tf, is to be added to the deduced latitude, or to the sum 
of the other corrections in order to obtain the latitude. 

The hour angle P, in seconds of time is known from 
t-}-E—a; t being the chronometer time of observation, E 
the chronometer error, and a the stars R. A. 

We therefore have the following complete formula for 
the latitude : 

d, + 6„ nin-m, (4 + r,) - (4 + I's) 
9 — — ^ — + ^ 2 4 

P dr 

+ ^{m,--m,)i^ ,- + [6.1347] P^ sin 2(^. 

See Form 9 b. The results of all the pairs are discussed by 
Least Squares. 

This method, although extremely simple in theory, in- 
volves considerable labor. It has, however, been employed 
almost exclusively on the Coast and other important Gov- 
ernment surveys, with results which compare favorably 
with those obtained by the first-class instruments of the 
fixed observatory. 

4. Latitude by Polaris off the Meridian. See Form 10.— 

This method depends upon the fact that the astronomical 
latitude of a place is equal to the altitude of the elevated 
pole. 



PRACTICAL ASTRONOMY. 61 

This latter is obtained by measuring the altitude of Polaris at 
a given instant, and from the data thus obtained, together with 
the star's polar distance, passing to the altitude of the pole. 

To explain this transformation: 
Let P= star's hour angle, measured from the upper meridian. 
a = altitude of star at instant P. corrected for refraction. 
d -— polar distance of star at instant P. 
qj = latitude of place. 
Then from the ZPS triangle we have 

sin a = sin cp cos d + cos cp sin d cos P. (107) 

This equation vrhich applies to any star may be solved 
directly; but with a circum-polar star it is nuich simpler to 
take advantage of its small polar distance, and obtain a 
development of cp in terms of the ascending powers of d, in 
which we may neglect those terms which can be shown to 
be unimportant. 

Now if we let x = the difference in altitude between Polaris 
at the time of observation and the pole, we shall have 

cp= {a — x). sin qj = sin (o — x), cos cp = cos (a — x), 
and from (107), 

1 = cos X (cos d + sin d cot a cos P) 
— sin x (cos c/ cot a — sin dcos P). ^ 

Moreover it is evident that if we can obtain the development 
of X in terms of the ascending powers of d, we will have 
the development of cp in the same terms, from (p = a — x. 
This is the end to be attained. Therefore let 

•r = Ad + B d' + C d^ + etc. , (109) 

be the undetermined development desired, in which A, B, C, 
etc., are to have such constant values, that the series, 
when.it is convergent, shall give the true value of x, what- 
ever may be the value of d. 

It is manifest then, that if this assumed value of x be 
substituted in (108), the resulting equation must be satisfied 
by every value of d which renders (109) convergent; that is, 
the resulting equation must be identical; otherwise (109) 
could not be true. 

With a view, therefore, to this substitution, let it be noted 
that by the Calculus, we have 



f>2 



PRACTICAL ASTRONOMY, 



COS X = 1 — 



sm X = X 



T 

6 



24 

of 
l20 



-etc.. 



— etc. 



and hence from (109), 
cos x = l — 






--.4i?d^ + etc. 



43 
sin X — Ad '\- Bd'^ -^ { C ~ ~ 



d^ + etc. 



Also, 



and 



cos d = 1 — 


cP d' 
2 +24"^^^ 


sin d^d — 


d' ^ d' 
6 +120"^* 



{m} 



in) 



11(1) 



111) 



(112) 



(113) 



Kow d is a very small angle; at present about 1° 16'. or 
0.0221 radians; x can never be greater than d, and in the 
general case will be less. Under these circumstances the 
above series becomes very convergent, and the sum of a few 
terms will represent with great accuracy the sum of the 
series. It is for this reason that the problem under discussion 
is applicable only to close circumpolar stars, and therefore, 
we take advantage of the small polar-distance of Polaris. 

Substituting (110), (111), (112), and (113), in (108). we have, 
rejecting terms involving the 4th and higher powers of d. 



A cot a 


2 


1- 5 cos P 


cos P cot a 


2 


A^ cos P cot a 



cP+ -! 



^4cosP 
B cot a 
.4- 



}d:' + 



-AB 



- C- 



cot a 



f cos P cot a ] 



— A cot ci 



PRACTICAL ASTRONOMY. 63 

This equation being identical, the algebraic sum of the co- 
efficients of each power of d must be separately equal to 
zero. 

Hence we have In' solution, 

A = cos P. 

,-, sin-P ^ 

/> = tan (K 

_ cos Psin^P 

Therefore, from (lOM). 

.r = d cos P — i d^ sin- P tan a + ^ cP cos P sin'^P. 
and 

(p = a — d cos P-\- i d' sin^ Ptan a — ^ d^ cos Psin^P + etc. . 

wliich is the required development. 

From (m), {n), (112). and (IKi). it is seen that d and .r are 
expressed in radians. 

Expressing them in seconds of arc. we have finally, since 
.'• and d in radians are equal to x and d in seconds multiplied 
l)y sin 1". 

cp = a ~ d cos P-\-id^ sin 1 " sin'^P tan n 

~ Jid-' sin- l"cos P sin^ P + etc. (1 U) 

The last three terms are in seconds. 

Hence we have the general rule: 

Take a series of altitudes of Polaris at any convenient time. 
Note the corresponding instants by a chronometer, prefer- 
ably sidereal, whose error is well determined. Correct each 
observed altitude for instrumental errors and refraction. 
Determine each hour angle by P = sidereal time — R. A. 

Take from the Ephemeris the star's polar distance at the 
time, being careful to use pp. 302 — 313. where also the R. A. 
required above will be found. 

Substitute each set of values in Equation (UJ:). and reduce 
each set separately. The mean of the resulting values of cp 
is the one adopted. See Form 10. 

As before stated, the method is applicable only to close 
circum-polar stars. Polaris is selected since it is the nearest 



64 PRACTICAL ASTRONOMY. 

bright star to the pole, a fact which is of importance in sex- 
tant observations. 

On the last page of the Ephemeris are given tabular values 
of the correction x. They are however only approximate: 
and the complete solution, as given above, consumes but very 
little more time. 

This is a very convenient method of determining latitude ; 
our only restriction being that, with a sextant, the observa- 
tions must be made at night. With the "Altazimuth" in- 
strument, the observations may be made for some time before 
dark. 

The last term in (114) is very small. In order to ascertain 
whether it is of any practical value, let us determine its 
maximum numerical value. = Denoting the term by z, and 
its constant factors by c, we have 

2 = c cos Psin^P. 

Eeplacing sin^Pby 1— cos^P, and differentiating twice, we 
have, after reduction 

dz 



dP 

d'z 



2 c sin P — 3 c sin^P. 

2 c cos P — 9 c sin^P cos P. 



dP' 
To obtain the maximum, 

2c sin P- 3c sin^P=0. 
From the roots of this we have 

sin^P=0, or sin^P=|, 

the last of which only, makes the second differential co- 
efficient negative. Hence the maximum value of the term 
is given when sin^P=f, or when z — id^sin^V^^Vi. For 
d = l°16', this gives 2; = . 2859". 

The maxrmum error committed by the omission of this 
term will therefore be about 0.3". Evidently its retention 
when the observations have been made with a sextant, would 
be superfluous. 

The value of log sin 1", not given in ordinary tables, is 
4.68557-10. 



PRACTICAL ASTRONOMY. 65 

Any mistake as to the value of P will manifestly produce 
its greatest effect when the star is moving- wholly in altitude. 
Hence if the chronometer error be not well determined, the 
times of elongation are the least advantageous for obser- 
vation. 

Since cos ( — P) = ccs P. we may measure P from the upper 
meridian to 180° in either direct loiu 

Latitude by Equal Altitudes of Two Stars. See Form 11.— 

By this method the latitude is found from the declinations 
and hour angles of tw^o stars; the hour angles being subject 
to the condition that they shall correspond to equal altitudes 
of the stars. 

Let 6 and 6' = the correct sidereal times of the observations. 
a and a' ~ the apparent right ascensions of the stars. 
^ and 6' — the apparent declinations of the stars. 
P and P' — the apparent hour angles of the stars. 
a = the common altitude. 

q) = the required latitude. 

P and P' are given from 

P=6-a. p'^O' - a'. 

From the Z P S triangle we have 

sin a — sin (p sin S + cos (p cos d cos P. 

sin a — sin cp sin 6' + cos cp cos S' cos P. 

Subtracting the first from the second and dividing by 
cos cp^ 

tan cp (sin S' — sin d) = cos S cos P— cos ^' cos P'. (115) 

The value of tan cp might be derived at once from this 
equation, since it is the only unknown quantity entering it. 
The form is, however, unsuited to logarithmic computation. 
In order to obtain a more convenient form, observe that the 
second member may be written 

'cos(JcosP cos (J' cos P'\ /cosScosP cos (J' cos P' 

+ 2 2 , 



Q6 PRACTICAL ASTRONOMY. 

Adding to the first parenthesis 

/cos 6 cos P' cos 6' cos P\ 
\ 2 2 J ' 

and subtracting the same from the second, we have, after 

factoring, 

tan cp (sin 6' — sin S) =i (cos S — cos (!>') (cos P -\- cos P') 
+ i (cos ()" + cos 6') (cos P — cos P'). 

Solving with reference to tan qj, and reducing by Formulas 
16, 17, and 18, Page 4, Book of Formulas, 

tan ^ == tan i {6' + d) cos i (P' + P^) cos ^{P' -P) . 

+ cot J ('(^ - 6) sin 1 (P' + P) sin J (P' - P). ^^^^^ 

The solution may be made even more simple by the use of 
two auxiliary quantities, m and M, such that 

m cos if = cos J (P' - P) tan J ((J' + (^) (117) 

?7^ .sin ilf= sin J (P'-P) cot i ((J'- ^) (118) 
Then 

tan cp = 771 cos [i {P' + P) -M]. (119) 

Equations (117) and (118) give m and M, and (119) gives 
q), all in the simplest manner. 

For example to find M, divide (118) by (117), and we 
obtain 

tan M^ tan i (P' - P) cot ^ {d' - d) cot J {S' + d). 

This admits of easy logarithmic solution. 

The value of m follows from either (117) or 118), and that 
of cp from (119), both by logarithms. 

The value of a does not enter; hence the resulting lati- 
tude will be entirely free from instrumental errors, those of 
graduation, eccentricity, and index error, and its accuracy 
will depend only upon the skill of the observer, and the 
accuracy of our assumed chronometer error and rate. 
Ephemeris stars should be chosen if possible, for the sake of 
accuracy in declinations, and their R. A. should permit the 
observations to be made with so short an interval that the 
refractive power of the atmosphere can not have changed 
materially in the mean time. The value of refraction is not 



PRACTICAL ASTRONOMY. 67 

required; it is only necessary that it remain practically 
constant. 

It may be shown that any error in observation or in the 
assumed chronometer correction will have its minimum 
effect upon the resulting latitude when the stars are so 
selected that the mean of their declinations is approximately 
the latitude, and also so that the observations can be made 
at nearly equal distances from the prime vertical, one north 
and the other south. 

Where several observations with the sextant are taken in 
succession on each star, it is better to reduce separately the 
pair corresponding to each altitude. 

LONGITUDE. 

The difference of Astronomical Longitude between two 
places is the spherical angle at the celestial pole included 
between their respective meridians. By the principles of 
Spherical Geometry, the measure of this angle is the arc of 
the equinoctial intercepted by its sides; or it is the same 
portion of 300" that this arc is of the whole great circle. 

But since the rotation of the earth upon its axis is per- 
fectly uniform, the time occupied by a star on the equinoc- 
tial in passing from one meridian to another, is the same 
portion of the time required for a complete circuit that the 
angle between the meridians is of 300°, or, that the inter- 
cepted arc is of the whole great circle. Moreover, all stars 
whatever their position occupy equal times in passing from 
one meridian to another due to the fact that all points on a 
given meridian have a constant angular velocity. 

The same facts apply also to the case of a body which, 
like the mean sun, has a proper motion, provided that motion 
be uniform and in the plane of, or parallel to, the equinoctial. 

Hence it is that Longitude is usually expressed in time; 
and in stating the difference of longitude between two 
places in time, it is immaterial whether we employ sidereal 
or mean solar time: for the number of mean solar time units 
required for the mean sun to pass from one meridian to 
another, is exactly equal to the number of sidereal time 
units required for a star to pass between the meridians. 



6S PRACTICAL ASTRONOMY. 

The astronomical problem of longitude consists, therefore, 
in determining the difference of local times, either sidereal 
or mean solar, which exist on two meridians at the same 
absolute instant. 

Since there is no natural origin of longitudes or circle of 
reference as there is in case of latitude, one may be chosen 
arbitrarily, and which is then called the "first" or "prime 
meridian." Different nations have made different selections: 
but the one most commonly used throughout the world is 
the upper meridian of Greenwich, England, although in the 
United States frequent reference is made to the meridian of 
Washington. 

The astronomical may differ slightly from the geodetic 
or geographical, longitude, for reasons given under the 
head of latitude. 

In the following pages, only the former is referred to; it is 
usually found from the difference of time existing on the 
two meridians at the instant of occurrence of some event, 
either celestial or terrestial. Up to about the year 1500 A. 
D. , the only method available was the observation of Lunar 
Eclipses. But with the publication of Ephemerides and the 
introduction of improved astronomical instruments, other 
and better methods have superseded this one, of which the 
two most accurate and most generally used, are the 
"■Method by Portable Chronometers," and the "Method by 
Electric Telegraph." Longitude may also be found from 
"Lunar Culminations" and "Lunar Distances," in cases 
when other modes are not available. 

1. By Portable Chronometers. Let A and B denote the 
two stations the difference of whose longitude is required. 
Let the chronometer error (E) be accurately determined for 
the chronometer time T, at one of the stations, say A; also 
its daily rate (r). 

Transport the chronometer to B, and let its error {E'} on 
local time be there accurately determined for the chronome- 
ter time T\ Let i denote the interval in chronometer days 
between T and T\ 

Then, if r has remained constant during the journey, the 



PRACTICAL ASTRONOMY. 69 

true local time at A corresponding to the chronometer time 
r will be, r + E+ir. 

The true time at B at the same instant is, T + E'. 

Their difference = difference of Longitude is 

X = E-}-ir- E\ (120) 

Thus the difference of Longitude is expressed as the differ- 
ence between the simultaneous errors of the same chronome- 
ter upon the local times of the two meridians, and the abso- 
lute indications of the chronometer do not enter except in 
so far as they may be required in determining i. 

The rule as to signs of E and /*, heretofore given, must be 
observed. If the result be positive, the second station is 
west of the first; if negative, east. 

This method is used almost exclusively at sea, except in 
voyages of several weeks, the chronometer error on Green- 
wich time, and its rate, being well determined at a port 
whose longitude is known. Time observations are then 
made with a sextant whenever desired during the voyage, 
and the longitude found as above. The same plan may evi- 
dently be followed in expeditions on land, although extreme 
accuracy cannot be obtained since a chronometer's "travel- 
ing rate '' is seldom exactly the same as when at rest. 

In the above discussion, the rate was found only at the 
initial station. If the rate be determined again upon reach- 
ing the final station, and be found to have changed to r', 

r + r' 
then it will be better to employ in the above equation — - — 

instead of r. To redetermine the longitude of any inter- 
mediate station in accordance with this additional data, we 

r'—r 
have .r= — . = dailv chanqe in rate; and the accumulated 

error at any station, reached n days after leaving A, would 

be £'+ (r + x~\n, the quantity in parenthesis being the 

rate at the middle instant. 

The above method is slightly inaccurate, since we have 
assumed that the chronometer rate as determined at one of 
the extreme stations (or both, if we apply the correction 



70 PRACTICAL ASTRONOMY. 

just explained), is its rate while en route. This is not as a 
rule strictly correct. 

Therefore, when the difference of longitude between two 
places is required to be found with great precision, "Chro- 
nometric Expeditions'' between the points are organized 
and conducted in such a manner as to determine this travel- 
ing rate. 

As before, 
let E =chron. error on local time at A at chron. time T. 

i i Tpf __ii a a a n ki a rpf 

" E^' = " " '' " B '' " T". 

ii rpfff—. a .< a k* A 4. a rpf n 

That is, the error on local time is determined at the first 
station for the time of departure, then at the second station 
for the time of arrival; again at the second station for the 
time of departure, and finally at the first station for the 
time of arrival. 

Then the entire change of error is E'" — E. But of this 
E" — E' accumulated while the chronometer was at rest at 
the second station. The entire time consumed was T"'—T. 
But of this T"~T' was not spent in traveling. .Therefore, 
the traveling rate, if it be assumed to be constant, will be 
{E'"-E)-{E''~E') 
' {T'" T) iT'' T'^ ' U'^W 

This, then, is the rate to be employed in Eq. (120) instead of 
the stationary rate there used. 

If the rate has not been constant, but -as is often the case, 
uniformly increasing or decreasing, the above value of r is 
the average rate for the whole traveling time of the two 
trips, whereas for use in Eq. (120), Ave require the average 
rate during the trip from A to B. This latter average will 
give a perfectly correct result provided the rate change iini- 
formly. If the rate has been increasing, then r in Eq. (121) 
will be too large numerically, by some quantity as x. Hence 
Eq. (120) becomes 

X:=zE + i{r-x)-E\ (122) 

in which r is found by (121.) In order to eliminate x, let 
the chronometer be transported from B to A, and return: 



m. PRACTICAL ASTRONOMY. 71 

/". e., take B instead of A as the initial point of a second 
journey. This is hest accomplished by utilizing the return 
trip of the journey ABA, as the first trip of the journey 
BAB. 

Then the new average rate r' having been found as be- 
fore, it will, if the trips and the interval of rest have been 
practically equal to those of the first journey, exceed the 
value required, by the same quantity, x, due to the vni- 
f or I nit y in the rate's change. Hence for this journey Eq. 
( 1-^0) becomes. 

A = E'" - [i{r' - .T) + E"l (123) 

In the mean of {Vl'l) and (123), .r disappears, giving. 

Hence, if our time observations are accurate, and the 
traveling rate constant, the difference of longitude between 
A and B may be determined by transporting the chro- 
nometer from A to B, and return. Or, if the rate be uni- 
formly increasing or decreasing, the difference of longitude 
will be found by transporting the chronometer from A to B, 
and return, then back to B: thus making three trips for the 
complete determination. 

In a complete " Chronometric Expedition," however, many 
chronometers, sometimes GO or 70, are used, to guard against 
accidental errors: and they are transported to and fro many 
times. As an example, in one determination of the longi- 
tude of Cambridge, Mass., with reference to Greenwich, 4:4= 
chronometers were employed, and during the progress of 
the whole expedition, more than 400 exchanges of chro- 
nometers were made. 

They are rated by comparison with the standard obser- 
vatory clocks at each station, which were in turn regulated 
by very elaborately reduced observations on, as near as 
possible, the same stars. 

Conducted as above described, ''Chronometric Expedi- 
tions " give exceedingly accurate results, especially if cor- 
rections be made for changes in temperature during the 
journeys. 



72 PRACTICAL ASTRONOMY. 

2. Longitude by the Electric Telegraph. See Fcrm 12. 

This method consists, in outline, in comparing the times 
which exist simultaneously on two meridians, by means of 
telegraphic signals. These signals are simply momentary 
^'breaks" in the electric circuit connecting the stations, the 
instants of sending and receiving which are registered upon 
a chronograph at each station. Each chronograph is in cir- 
cuit with a chronometer which, by breaking the circuit at 
regular intervals, gives a time scale upon the chronograph 
sheet, from which the instants of sending and receiving are 
read off with great precision. 

Suppose a signal to be made at the eastern station (A) at 
the time T by the clock at A, which signal is registered at 
the western station (B) at the time T' by the clock at B, 

Then if E and E' are the respective clock errors, each on 
its own local time; and if the signals were recorded instantly 
at B, then the difference of longitude would be (T+E) — 
{T' + E'). But it has been found in practice that there is 
always a loss of time in transmitting electric signals. 
Therefore in the above expression {T' + E') does not corre- 
spond to the instant of sending the signal, but to a some- 
what later instant. It is therefore too large, the entire 
expression is too small, and must be corrected by just the 
loss of time referred to. This is usually termed the "Re- 
tardation of Signals;" and if it be denoted by x, the true diff- 
erence of longitude will be {T+E)-{r + E')^x = V -vx = l. 
But X is unknown, and must therefore be eliminated. 

In order to do this, let a signal be sent from the western 
station at the time T" w^hich is reached at the eastern at 
the time T". Then if E" and E'" are the new clock errors, 
the true difference of longitude will be 

(T'" + E'") - {T" -vE")-x^\" -x^\. 

By addition, x disappears, and if A denote the longitude, we 
will have 



Or, in full, assuming that the errors do not change in the 
interval between signals, 



PRACTICAL ASTRONOMY. 73 

^ ^ [( T-r)-\ -(E -E')'[+[{T"'-r') + (E-E')] 

T, T, T" and T"\ are given by the chronograph sheets: 
E and E' must be determined with extreme accuracy, since 
incorrect values will affect the resulting longitude directly. 

Having established telegraphic communications between 
the two observatories (field or permanent), usually by a 
simple loop in an existing line, preliminaries as to number 
of signals, time of sending them, intervals, calls, precedence 
in sending, etc., are settled. The electric apparatus for this 
purpose is arranged in a simple circuit consisting of one or 
at most two galvanic cells, a break-circuit key, chronograph, 
and break -circuit chronometer. At about nightfall messages 
are exchanged as to the suitabilit}^ of the night for obser- 
vations at the two stations. If suitable at both, each observer 
makes a series of star observations with the transit to 
find his chronometer error. This series should involve at 
least ten well determined Ephemeris stars, three equatorial 
and two circumpolar for each position of the instrument. 
As the time agreed upon for the exchange of signals 
approaches, the local circuit should be connected by a relay 
to the main line, which is worked by its own permanent 
batteries, and in which there is also a break-circuit key. 
By this arrangement it is seen that each chronogrciph will 
feceive the time-record of its own chronometer; and cdso 
the record of any signals sent over the main line in either 
direction. 

Neither chronograph receives the record of the other's 
chronometer. Then at the time agreed upon, warning is 
sent by the station having precedence, and the signals fol- 
low according to any prearranged system. Notice being 
given of their completion, the second station signals in the 
same manner. 

As an example of a system, let the break-circuit key in 
the main line be pressed for 2 or 3 seconds once in about 
ten seconds; this being continued for five minutes will give 
:^1 arbitrary signals from each station. 

Each chronometer sheet when marked with the date, one 
or more references to actual chronometer time, and the 



74 PRACTICAL ASTRONOMY. 

error of chronometer, as soon as found, will, in connection 
with the sheet from the other station, afford the obvious 
means of finding all the quantities in Eq. (125) from which 
the longitude is computed. The sheets may be compared 
by telegraph, if desired. 

The work of a single night is then completed by transit 
observations upon at least ten more stars under the same 
conditions as before, the entire series of twenty being so 
reduced as to give the chronometer error at the middle of 
the interval occupied in exchanging signals. The mode of 
making this reduction will be explained hereafter. 

Of course this method involves the error of supposing the 
unknown x to be constant, which may not be quite the case, 
since the times required to demagnetize the relays at the 
two stations may be slightly unequal. This error will be 
exceedingly small, and may be found by a comparison of 
the two relays before or after the completion of the work 
on the same principle that the "personal equation" between 
two observers is found. It may be however and is frequently, 
reduced by one-half as follows : A station in sending, throws 
its chronometer into the main line, that of the receiving 
station being still in the local circuit. In this way the 
chronograph at the latter station registers the beats of eacl} 
chronometer, and the one at the former station, the beats 
of its own chronometer only. Each registers any arbitrary 
signals that may be sent, although in this case none are 
necessary, since the beats of the sending chronometer sup- 
ply the deficiency. This operation having been performed 
at each station, we evidently have upon each chronograph 
sheet the comparison of the two clocks; and a communica- 
tion giving the clock error and the local time of the first or 
last full minute beats, or better still, the chronograph 
sheet itself, affords the obvious means of solving Eq. (125). 

This method although slightly more accurate in theory 
than the former, involves the printing of both time records 
simultaneously on the same sheet, and the consequent inac- 
curacies in reading which might occur, due to a close but 
not perfect, coincidence in beats. 



PRACTICAL ASTRONOMY. 75 

The final adopted value of the longitude should be the 
mean of the results of at least five nights. 

Reduction of the Time Observations. See Form 12 a. — These 
observations, as just stated, are in two groups; one before, 
and one after the exchange of signals or comparison of 
chronometers. From them is to be obtained the chronome- 
ter error at the epoch of exchange or comparison, which is 
assumed to be the middle of the interval consumed in the 
exchange: this latter being about 12 minutes. 

Let us resume the equation of the Transit Instrument 
approximately in the meridian, 

a=T + E -^ a A -{-bB + C(c-. 021 cos cp), (12G) 

and let Tl, denote the epoch, or the known chronometer time 
to which the observations are to be reduced. Let us suppose 
also, that of the three instrumental errors a, b, and c, only 
b has been determined, this being found directly by reading 
the level for every star. The rate of the chronometer, r, is 
supposed to be known approximately, and it is to be borne 
in mind that E is the error at the time T. Then in the above 
equation E, a, and c, are unknown. 

Now if we denote the error at the epoch by E^, we shall have 

E=E,-(T,-T)r. (127) 

And if E'o denote an assumed approximate value of Eq, and 
e be the unknown error committed by this assumption, we 
shall have, 

E=Wo + e-{T,-T)r. (128) 

From which Eq. (120) becomes, 
f + .-la+ Cc^ T-.021 cos cpCi- E\- (T,-T) r + Bb - a =0, 

in which everything is known save f (the correction to be 
applied to the assumed chronometer errors at the epoch), 
a, and c. 

Aa is called the correction for azimuth. 

Cc " " " " collimation. 

— .021 cos ^C " " " " diurnal aberratn. 

-(To-T)r •' " " " rate. 

Bb '' '' " '' level. 



76 PRACTICAL ASTRONOMY. 

Collecting the known terms, transposing them to the 2d 
member, and denoting the sum by n, we have, 

e + Aa+ Cc=^n. (129) 

Each one of the twenty stars furnishes an Equation of 
Condition of this form, from which, by the principles of 
Least Squares we form the three "Normal Equations," 

2{A)e + 2 (A') a + 2(AC)c = 2{An), 
^(l)f + ^ {A) a + :2 (C) c = :S(n), 
from a solution of which we find a, c, and the correction, f , 
to be applied to the assumed chronometer error at the 
epoch. 

If either c or a be known, say c, by methods given under 
"Time by Meridian Transits," then the correction for colli- 
mation for each star, Cc, should be transferred to the 
2d member and included in n. We then have only the two 
"Normal Equations," 

^(1) e + 2{Aa) ^:S{n), 
:^ (A) e + 2 (A'^) a = 2 {An), 
from which to find e and a. 

It is to be remembered that the middle ten stars have been 
observed with the instrument reversed, and that such reversal 
changes the sign of c, and therefore of the term Cc. Hence 
in forming the "Equations of Condition" for those stars, 
care should be taken to introduce this change by reversing 
the sign of C. The sign of c as found from the "Normal 
Equations " will then belong to the collimation error c of 
the unreversed instrument. 

Also, since reversing the instrument almost invariably 
changes a, it is better to write a' for a in the correspondmg 
"Equations of Condition," and treat a' as another unknown 
quantity. We will thus have four "Normal Equations" in- 
stead of three, and derive from them two values of the azi- 
muth error, one for each position of the instrument. 

Sometimes, and perhaps with even greater accuracy, the 
solution is modified as follows: 

Independent determination of a and c, are made, as ex- 
plained heretofore, by the use of three stars. 



PRACTICAL ASTRONOMY. 77 

With these, correcting also for rate, each star gives a value 
of the chronometer error at the epoch as shown in Form 1. 
The principle of Least Squares is then applied in the 
manner just detailed, to obtain the corrections to be applied 
to these values of a. c, and the mean chronometer error. 
With these corrected values of a and c, new values of the 
chronometer error are found by direct solution (Form 1), the 
mean of which is adopted. 

"Longitude by the Electric Telegraph" had its origin in 
the U. S. Coast Survey, and has since been employed con- 
siderably in Europe. As at first employed it consisted virtu- 
ally in telegraphing to a western, the instant of a fixed star's 
culmination at an eastern station: and afterwards, telegraph- 
ing to the eastern, at the instant of the same star's culmi- 
nation at the western station. 

In connection with Talcott's Method for Latitude, it has 
been used extensively in important Government Surveys, 
taking precedence, whenever available, over all other 
methods. 

3. Longitude by Lunar Culminations. — The moon has a 
rapid motion in Right Ascension. If. therefore, we can find 
the local times existing on two meridians, when the moon 
had a certain R. A., their difference of longitude becomes 
known from this difference of times. 

Determine the local sidereal time of transit or R. A., of 
the moon's bright limb, and denote it by ai. 

From pp. 385—392, Ephemeris, take out the R. A. of the 
center at the nearest Washington culmination. This ± Side- 
real Time of semi-diameter crossing the meridian, according 
as the east or west limb is bright, taken from same page, 
will give the R. A. of the bright limb, at its culmination at 
Washington. Denote this by ^,v. 

Now if an approximate longitude be ncA known, which 
will seldom be the case, one may be established as follows: 
Let V = moon's change in R. A. for one hour of longitude, 
taken from same page of Ephemeris. Then upon the sup- 
position that this is uniform, we will have 

V : \::ai — a,,. : L\ or L = — , 



78 PRACTICAL ASTRONOMY. 

L^ being the approximate longitude from Washington, whose 
longitude from Greenwich is accurately known. With this 
value of L' take from the Ephemeris a new value of r 
corresponding to the mid-longitude \ L', and determine as 
before a closer approximate longitude, U\ If we are within 
two hours of Washington in longitude, L'^ will be sufficiently 
close /or the purpose to tvhich we are to apply it. If farther 
away, make one or two more approximations, and call the 
final result Lap. 

Lap will be true within a very few minutes of time even if 
the observing station be in Alaska, situated 6 hours from 
Washington, and even if the observations be made when 
the moon's irregularities in R. A. are most marked. 

With the approximate longitude (and this is the use to be 
made of this quantity, before referred to), we may now find 
the sidereal time required for moon's semi-diameter to 
cross the meridian of the place of observation by simple 
interpolation to 2d or 3d differences in the proper column of 
the same page of the Ephemeris. Denote this by Ti. Perform 
this interpolation with care. 

The greatest change in the time required for semi-diameter 
to cross the meridian, due to a change of one hour in longi- 
tude, is about O.IS^^''. Hence, even if we could possibly have 
made an error of 10 minutes in our determination of Lap, the 
value of Ti can only involve an error of about .OS^""'' tvhen at 
its maximum. This would involve a maximum error of 
about 0.5''''*' in the resulting longitude. 

ai ± Ti = a^ will then be the R. A. of the moon's center at 
the instant of transit of the center. 

On Pages V to XII of the Monthly Calendar are found the 
R. A. of the moon's center for each hour of Greenwich mean 
time. The problem now is to find at what instant Tg of 
Greenwich time the moon's center had the R. A. determined 
by our observation. This may be solved by an inverse inter- 
polation; i. e., instead of interpolating a R. A. correspond- 
ing to a given time not in the table, we are to interpolate a 
time to a given R. A. not in the table; and in this interpola- 
tion the use of second differences will be quite sufficient. 



PRACTICAL ASTROXOMY. 79 

Therefore let 7',, and To + 1 be the two Greenwich hours 
between which a,, occurs. 

Let 6 a be the increase of moon's R. A. in one minute of 
mean time, at J\. This is given on the same page. 

Let 6' (X be the increase oi S a in one hour. Found from 
same column by subtracting adjacent values of da. 

Let a^ be the R. A. of the Ephemeris at Ji,. 

Then using second differences, we have 

In this equation T,j — T^, is expressed in seconds; every- 
thing is known but if, and its value may be found by a so- 
lution of the quadratic. The result added to To gives Tg, or 
the Greenwich meet it time at which the moon's center had 
a,, for its R. A. Convert this into Greenwich sidereal time, 
call the result a^j, and our longitude is known from 

X = a,j-a^., (131) 

The preceding is the method to be followed where there is 
but a single station. 

Imperfections in the Lunar Tables from which the Ephem- 
eris is computed, render the tabular R. A. liable to slight 
errors. Therefore from Equation (130) our values of Ty and 
hence a,, may be incorrect from this cause, giving from 
Equation (131) an incorrect longitude. 

Differences between two tabular values, are however 
nearly correct. 

Hence it is more accurate to have corresponding obser- 
vations taken for the transit of the same day at a station 
whose longitude is known. 

Its longitude, fovncl as above, will be 

and the difference of longitude between the two stations, 

inaccuracies of the Ephemeris being nearly eliminated in 
the difference (cyg' — cKg). 

No method of determining longitude by Lunar Culmina- 
tions is sufficiently accurate for a fixed observatory. It 



80 PRACTICAL ASTRONOMY. 

may however be used in surveys and expeditions where 
telegraphic connection with a known meridian can not be 
secured. Even with the appliances of a fixed observatory, 
the mean of several determinations is sometimes subse- 
quently found to be in error by from 4 to 6 minutes of time 
(Madras Observatory). Dependence should not therefore be 
placed in a single observation, but the operation should be 
repeated as many times as may seem desirable. The longi- 
tude derived from any determination may be employed as 
the approximate longitude required in any subsequent 
determination. 

Before proceeding to any details as to the observations 
and reductions, it is well to note the effect of errors in either, 
upon our result. The main outline of the problem consists 
in determining the moon's R. A. at a certain instant, and 
then ascertaining from the Ephemeris the Greenwich time 
of the same instant. Both the moon's R. A. and the instant 
are denoted, at the place of observation, by a^ — ai±Ti. 
(Xi depends very largely upon accuracy of observation and 
reduction. Ti depends upon interpolation with an approxi- 
mate longitude. As shown before, no error of assumed 
longitude that could ever occur in practice, would have any 
appreciable effect on Ti. If the interpolation be properly 
performed, Ti can involve only very slight errors. But what- 
ever they may be, they enter with full effect in a^, and when 
the final operation is performed to determine the correspond- 
ing Greenwich time, an inspection of the tables will show 
that any error in a^ is increased from 20 to 30 times in the 
resulting longitude. In this way, as before shown, an error 
of ,02^ in Ti is amplified into .5^ in the result. 

Errors in (Xi affect a^, and therefore the result in the same 
manner; hence we see that considerable care is necessary 
in both observation and reduction. At the very best, the 
result is liable to be in error from 1 to 3 seconds. In latitude 
of West Point, 1 second of time = 1142 feet in longitude. 

Observations and Reductions.— The transit instrument is 
supposed to be pretty accurately adjusted to the meridian, 
and the outstanding small errors a, h, and c, measured. The 
error and rate of the sidereal chronometer are so well known 



PRACTICAL ASTRONOMY. 81 

that the former may be found for any instant. The mode of 
reducing the effect of inaccuracies in these data will be ex- 
plained later. 

Note the chronometer time of transit of the moon's bright 
limb over each wire of the instrument. In this case, as with 
a star, the true time of transit is found by reducing the ob- 
servations to the middle wire and then correcting for the 
three instrumental errors. See Form 1. But in case of the 
moon these reductions and corrections take a somewhat 
modified form due to the two facts that the moon has a 
proper motion in R. A. , and also a very sensible parallax in 

R. A. when on a side wire. Hence we have — F instead of 

n 

sec 6'. for the reduction to the middle wire and (Aa-\-Bb 
n 

+ Cc') F cos ^' instead of A a + Bb-{-Cc, for the instrumental 
correction. (6' denotes the declination as affected by paral- 
lax, which in case of a star is identical with the geocentric 
declination, d, and may be written for it. ) In these expressions 

i^ = [1 - p sin TT cos {cp' - 6)] ^Q -^g^3 '_ '^^^^ sec d; p being 

the radius at place of observation in terms of the equatorial 
radius, tt the moon's equatorial horizontal parallax, cp' the 
central latitude, ^ the geocentric declination, and {S a) as 
already explained. These quantities must be determined be- 
fore the reductions can be made. The method of finding 
p and q/ has already been explained. To find tt, d, and (^«), 
take the recorded time of transit over the middle wire, apply 
the chronometer correction, and determine an approximate 
longitude as explained at the beginning of the method. With 
this. 7r may be taken from page TV, and S and {d a) from 
pp. V to XII, Monthly Calendar. F thus becomes known. 
6' = d — 7t sin (cp — 6). 

We have now all the quantities for determining the true 
sidereal time of transit of the limb over the meridian, or the 
R. A. of the limb at this instant; viz. : 

a,= —+E+ — F+{Aa + Bb+Cc') FcosS\ (132) 
The computation proceeds as already indicated. 



82 PRACTICAL ASTRONOMY. 

One of the greatest inaccuracies to be apprehended is a 
failure to determine a very exact value of E for the instant 
of transit. This quantity may be eliminated, or very nearly 
so, as follows: 

If two or more fundamental stars, those whose places have 
been established with the highest degree of accuracy, be 
selected so that the mean of the times of their transits shall 
be very closely the time of transit of the moon's limb, then 
the mean of their equations will be, corresponding to a mean 
star, 

«',--= ^- + £-, + ^ sec d\ +{Aa+ Bh+ Cc'),. (133) 

Subtracting from Eq. (132), since E and Eg denote errors 
at almost the same instant, we have 

2 T 2 T 2 i ^ i 

ai = a,+ ^ — ' + — F sec S, + etc., (134) 

u n n n 

in which E has disappeared. 

If E and Eg differ, their difference will be simply the 
change of error in, for example, ten minutes, which can be 
accurately allowed for by the chronometer's well established 
rate. Moreover, if the stars be selected so that their decli- 
nations differ but slightly. from that of the moon, it is evident 
that the last terms of Eqs. (132) and (133), will be nearly the 
same, and that their difference in Eq. (134), will be a mini- 
mum. See expressions for A, B, and C, in connection with 
Form 1. 

By this method, therefore, the R. A. of the moon's limb, 
ai, is from Eq. (134), made to depend very largely upon the 
R. A. of fundamental stars; instrumental and clock errors 
being reduced to a minimum of effect. 

The stars should be selected from the Ephemeris in accord- 
ance with the above conditions, and observed in connection 
with the moon. 

4. Longitude by Lunar Distances.— On pp. XIII to XVIII 

of the Monthly Calendar in the Ephemeris are found the true 
or geocentric distances of the moon's center from certain 
fixed stars, planets, and the sun's center, at intervals of 



PRACTICAL ASTRONOMY. 83 

3 hours Greenwich mean time. If then an observer on any 
other meridian determine by observation one of these 
distances, and note the local mean time at the instant, he 
can by interpolation determine the Greenicich mean time 
vv^hen the moon had this distance, and hence the longitude 
from the difference of times. 

The planets employed are Venus, Mars. Jupiter and Saturn,^ 
and the fixed stars, known as the 9 lunar-distance stars, are 
(X Arietis (Hamal), (X Tauri (Aldebaran), fi Geminorum 
(Pollux). (X Leonis (Regulus). (x Yirginis (Spica), ex Scorpii 
(Antares), ^r Aquilae (Altair). ^f Piscis Australis (Fomalhaut), 
and (v Pegasi (Markab). From this list, the object is so 
selected that the observed distance shall not be much less 
than 45°, although a less distance may be used if necessary. 

The distance observed is that of the moon's bright limb 
from a star, from the estimated center of a planet, or from 
the nearest limb of the sun. If the sextant telescope be 
sufficiently powerful to give a well defined disc, we may 
measure to the nearest limb of the planet, and treat the ob- 
servation as in the case of the sun. 

Thus in Fig. 7, letting Z represent the observer's zenith, 
and C" and C" the observed places of the sun and moon 
respectively, the distance measured is 8' J/', from limb to 
limb. 

The effect of refraction is to make an object appear too 
high, and that of parallax, too low. In the case of the sun 
the former outweighs the latter. In the case of the moon 
the reverse is true. Hence the true or geocentric places of 
the two bodies would be represented relatively by S and M, 
and the distance S i)/, from center to center, is the one 
desired. 

The outline of the method is as follows: 

Having measured the distance S' M'. and corrected it for 
the two semi-diameters ; and having also measured the alti- 
tudes of the two lower limbs and corrected them for the 
respective semi-diameters, we have in the triangle Z C C" 
the three sides given, from which we find the angle at Z. 
Then having corrected the observed altitudes for refraction, 
semi-diameter and parallax, we have in the triangle Z S M, 



84 PRACTICAL ASTRONOMY. 

two sides and the included angle Z, to compute the opposite 
side 8M, 

Before proceeding to the more definite solution, three 
points should be noticed. 

1st. The semi-diameter of the moon as seen from the sur- 
face of the earth is greater than it would appear if measured 
from the center of the earth, due to its less distance. Hence 
CM' is an ''augmented semi-diamete7^" and must be 
treated accordingly. The augmentation in case of the sun is 
insignificant. 

2d. Since refraction decreases with the altitude, the re- 
fraction for the center of the sun or moon will be greater 
than that for the upper limb, and that of the lower limb will 
be greater than that of the center. The apparent distance 
of the limbs is therefore diminished, and the whole disc, in- 
stead of being circular, presents an oval figure, whose verti- 
cal diameter is the least, and horizontal diameter the grea^test. 
as shown in Fig. 8. Therefore if c d denote the direction 
of the measured distance, the assumed semi-diameter, c f\ 
will be in excess by the amount e /, and must be corrected 
accordingly. This correction becomes of importance if the 
altitude of either sun or moon be less than 50° at the moment 
of observation. 

3d. Since the vertical line at the station does not in general 
pass through the earth's center, but intersects the axis at 
a point R, (see Fig. 5,) it is most convenient to reduce our 
observations at first to the point i?, regarding the earth as 
a sphere with i? O as a radius, and then to apply the small 
correction due to the distance C R, in order to pass to the 
true or geocentric quantities. 

In the following explanation, the body whose distance 
from the moon is measured, is taken to be the sun. The result 
will then apply equally to a planet if its limb be considered; 
if its center be considered, the expression for its semi- 
diameter becomes zero. If the body be a fixed star, the 
expressions for its semi-diameter and parallax become zero. 
Let /?/' = apparent altitude of moon's lower limb. 
H''= apparent altitude of sun's lower limb. 



PRACTICAL ASTRONOMY. 85 

d" ~ observed distance between nearest limbs of sun and 
moon, corrected for index error and eccentricity. 
T = local mean solar time at instant of measuring d". 
U = an assumed approximate longitude. 
(p = latitude. 
Note the readings of the barometer and of the attached 
and external thermometers. 

With T and L% take from the Ephemeris the following 
quantities : 

,s=geocentric semi-diameter of moon, from page IV, monthly calendar. 
;r=equatorial liorizontul parallax of moon, from page IV, monthly calendar. 
(5=rgeocentric declination of moon, from page V to XII, monthly calendar. 
/S^ semi-diameter of sun, from page I, monthly calendar. 
P=eq. hor. Parallax of sun, from page 278, Ephemeris. 
Z) = geocentric declination of sun, from page I, monthly calendar. 

We must now correct d" for both semi-diameters, aug- 
mented in case of the moon. Therefore with h/^-{-s and s 
as arguments, enter the proper table and take out the amount 
of augmentation. In absence of tables, this may be com- 
puted by 

Augmentation = A-s'-^cos (/i" + 5) + ^k''s^ + ^k^s^ cos' {h" + 5), 
in which log. A= 0.2405-10, and s is expressed in seconds. 

Add this correction to s and we have s'=moon"s semi- 
diameter as seen from point of observation. 

We now have (neglecting the distortion of discs), the fol- 
lowing values of the observed quantities reduced to the 
centers of the observed bodies, viz. : 

d' = d" + s' + S. h' = h" + s\ H' = H" + S. 

Using these quantities we may now find the correction due 
to distortion of discs (or refractive distortion), as follows: 
From tables of mean refraction take out the refractions cor- 
responding to the altitude (Z/' + s') of the upper limb, to that 
(h'—s') of the lower, and that (h') of the center. The differ- 
ence between the latter and each of the other two gives very 
nearly the contraction of the upper and lower semi-diame- 
ters of the moon. This may be repeated once if the refrac- 
tions are very great due to a small altitude. The mean of 
the two is the contraction of the vertical semi-diameter due 



86 PRACTICAL ASTRONOMY. 

to refraction. Denote it by A ,s, and the same quantity in 
case of the sun by A S. 

These quantities are represented by a b in Fig. 8, and from 
them we are to find ef, or the distortion in the direction of 
d'\ This is found to vary very nearly as cos^ q, q being the 
angle which d" makes with the vertical. 

The values of q and Q, in case of the sun, yAU be found 
from the three sides of the triangle Z C C" , Fig. 7. Their val- 
ues, page 6, Book of Formulas, will be, if m^\{d' -th' ^H'). 

. .,, cos 7^? sin (m— JT') . . 1 ^ cos m sin (m—/?/) 

sm' \q— -. .> ,, — '-, sm' \Q- — ; ^-^ — =^ . 

sm a cos h sm a cos H 

And the refractive distortions will be, from the above, 

Ascos^g, and a>Scos^§. 
Hence the fully corrected values of the measured quanti- 
ties are 

d'—d" + (s'- ascos^q) + (/S- A>S'cos'(3), 

We now have the distance (d'), between the centers and 
the altitudes of the centers (h' and H'), as these quantities 
would have been had we been able to measure them directly. 
We must now ascertain what they would have been, had 
we measured them at the center of the earth; or as a first 
step, had we measured them at the point R. 

Therefore let H^, tt^, h^, and d^, be the values of H\ n, h\ 
and d', when referred to E, and let r and r^ be the actual 
refractions for /^' and H\ P is not sensibly altered by refer- 
ence to point R. 

7t 

It may be shown that n ^ = — , very nearly and with even 

greater precision than is necessary for the present problem. 
Then, since in this first step the earth is regarded as a 
sphere, we have (Art. 82, Young), 

h^ = /?/— r + TT^ cos ( h'— r ). 
H] ^H'-r^ + P cos {W - r,) . 

In order to find d^, let S and M in Fig. 7 represent the 
places of the sun and the moon referred to the point R, and 



PRACTICAL ASTRONOMY. 87 

(^' and C", the same before such reference. Then in 
triangle ZC C" . 

„ cos c/'— sin/?' sin ii' ^ .. -d i ^ -r. i 

cos Z— Yi TT, • "age 6. Joook of Formulas. 

cos h' cos H' ^ 

From triangle Z S M. 

^ cos d,— sin/?, sin iT, ^ .. id i ^ t:^ i 

cos Z= ' — J —7 -'. Page h. Book of Formulas. 

cos n^ cos H^ 

Equating these two values of cos Z, adding unity to both 
members and reducing. 

c os cV + co s (/?/ + H') _ cos c/, + cos (/?, + i?J p 
cos/?' cos il' ~ cos k^ cos H^ ' ^ 

Make ?>/=i(/?' + iJ' -^cZ'). whence cos (/?' + ^')=cos(2m-(7'). 
Substituting in the preceding equation reducing the first 
member by formulas 4, page 4, 11. page 2. and 13, page 1, 
and the second member by formulas and 10. page 2. we 
have 

cos m cos (ni — d') _ cos^i(/i^ + jffJ — sin^d^ 
cos/?' cos i?' cos/?, cos ^, 

Whence. 

•217 01,7 , TT\ cos/?, cos H. , ,,, 

8mH(:/,=cos-4^(/?,+iZ,) ~ ~ cos m cos im—d'). 

" ' " ' ' " cos /? cos if ^ ^ 

This may be placed in a more convenient form by assuming- 
cos /?, cos iJ, cos /?? cos (???. — d') _ . .iiir 

cos/?' cos Jf' cosH (/?, + J?,) ~ " 
Whence sin i d, = cos ^ {h^ + H^) cos 31. 

We now have the distance between the centers as it would 
have been without refraction, if measured from the point B. 

The small correction needed to reduce this to the earth's 
center is, 

. /sinD sin d 



n I 



\sin d, sin d. 



where i= — =^ \\<p being the latitude, and e the 

S/\ — e^ sin^^ 

eccentricity of the meridian = .0816967. 



88 PRACTICAL ASTRONOMY. 

Hence we have finally, denoting the geocentric distance 
between centers by cl, 

, , . / sin D sin S 

d — d^^ni 



sin d^ sind^. 

This operation of finding cl from the observed quantities is 
called ''Clearing the Distance/' 

It is now necessary to find the Greenwich mean time 
when the moon and sun were separated by the distance d. 
For this purpose, enter the Ephemeris at the pages before 
referred to, and find therein two distances between which d 
falls. Take out the nearer of these and the Greenwich hours 
at the head of the same column. Then if A denote the differ- 
ence between the two distances, and a' the difference 
between the nearer one and d, both in seconds, we shall 
have, using only first differences, for the correction, f, to 
be applied to the tabular time taken out. 






A:3^:: A':f^ . '. f^= -^ A 
A 

Or, log f ^ = log — + log A '. 
Or m seconds, log r = log \- log A . 

The logarithms of are given in the columns headed 

"P. L, of Diff.^' (Proportional Logarithm of Difference.) 
Hence we have simply to add the common logarithm of A ' 
in seconds to the proportional logarithm of the table to obtain 
the common logarithm of the correction in seconds of time. 

To take account of second differences, take half the differ- 
ence between the preceding and following proportional loga- 
rithms. With this and t as arguments enter Table 1, Ap- 
pendix to Ephemeris, and take out the corresponding seconds, 
which are to be added to the time before found when the 
proportional logarithms are decreasing, and subtracted when 
they are increasing. 

Denote the final result by Tg, and the difference of longi- 
tude by A. Then 

\^Tg-T. (135) 



PRACTICAL ASTRONOMY. 89 

The mode given above for clearing the distance is quite 
exact but somewhat laborious. There are. however, several 
approximative solutions, readily understood from the fore- 
going, wliich may be employed where an accurate result is 
not required, and which may be found in any work on 
Navigation. 

The method by "Lunar Distances '* is of great use in long 
voyages at sea or in expeditions by land, where no meridian 
instruments are available, and when the rate of the chro- 
nometers can no longer be relied upon. 

It is important to note that if T in Eq. (135), denote the 
i'hrononieter time of observation, instead of the true local 
time. T,j— T will be the error of the chronometer on Green- 
wich time. In this way chronometers may be •'checked." 
If. however, T denote the true local time, obtained by apply- 
ing the error on local time to the chronometer time, then the 
same equation gives the longitude. 

Observations.— It is necessary that h" . H". and d" should 
correspond to the same instant T. Hence observe the follow- 
ing order in making observations. Take an altitude of the 
sun's limb, then an altitude of the moon's limb, then the 
distance, carefully noting tlie time, then an altitude of the 
moon's limb, then an altitude of the sun's limb. A mean of 
the respective altitudes of the two limbs will give very nearly 
the altitudes at the instant of measuring the distance. If 
neither body be near the meridian, two or three like groups of 
observations may be made. If the altitudes be not measured, 
they may be computed from the Z PS triangle, knowing the 
declination, latitude and hour angle. 

The accuracy of the result will depend upon the observer's 
skill ^vith the sextant, and mode of reduction follow^ed. 

OTHER METHODS OF DETERMINING LONGITUDE. 

1st. If two stations are so near each other that a signal 
made at either, or at an intermediate point, can be observed 
at both, the time may be noted simultaneously by the chro- 
nometers at the two stations, and the difference of longi- 
tude thus deduced. An application of the same system, by 
means of a connected chain of signal stations, will give 



90 PRACTICAL ASTRONOMY. 

the difference of longitude between two remote stations. 
The signals are usually flashes of light — either reflected sun- 
light or the electric light, passed through a suitable lens. 

2d. By noting the time of beginning or ending of a lunar 
or solar eclipse, or by occulations of stars by the moon. For 
these methods, see various Treatises on Astronomy. 

3d. By Jupiter's Satellites: a. From their eclipses. The 
Washington mean times of the disappearance of each satel- 
lite in the shadow of the planet, and reappearance of the 
same, are accurately given in the Ephemeris, pp. 452-473, 
accompanied by diagrams of conflguration for convenience 
of reference. A full explanation of the diagrams is given 
on p. 449. An observer who has noted one of these events, 
has only to take the difference between his own local time 
of observation and that given in the Ephemeris, to obtain 
his longitude. This method is defective since a satellite has 
a sensible diameter and does not disappear or reappear in- 
stantaneously. The more powerful the telescope employed, 
the longer will it continue to show the satellite after the 
first perceptible loss of light. These facts give rise to dis- 
crepancies between the results of different observers, and 
even between those of the same observer with different 
instruments. Both the disappearance and reappearance 
should therefore be noted by the same person with the same 
instrument and a mean of the results adopted. The first 
satellite is to be preferred, as its eclipses occur more fre- 
quently and more suddenly. 

h. From their occulations by the body of the planet. The 
times of disappearance and reappearance to the nearest 
minute only, are given on same pages of Ephemeris. Since 
the times are only approximate, they simply serve to enable 
two observers on different meridians to direct their attention 
to the phenomenon at the proper moment. A comparison 
of their times will then give their relative longitude. 

c. Fi^om their transits over Jupiter's disc. 

d. From tlie transits of their shadows over Jupiter's disc. 
The approximate times of ingress and egress, to be used as 
in case b, are given on same pages of the Ephemeris, for 
cases c and d. 



PRACTICAL ASTRONOMY. 91 

TIME OF MERIDIAX PASSAGE. 

To determine the local mean solar time of a given body 
coming to the meridian, it is to be noted that this time, (P), 
is simply the hour angle of the mean sun at that instant, 
and that this hour angle is. by the general formula. P= side- 
real time — R. A. of mean sun. 

Xow the sidereal time at the instant is equal to the R. A. 
of the body on meridian, and this is equal to its R. A. at the 
preceding Greenwich mean noon (a) plus its increase of R. 
A. since that epoch, which is equal to m (P+A), A being the 
longitude from Greenwich, and m the body's hourly in- 
crease in R. A. Or. sidereal tiine=a+m(P-\-X). 

Similarly vre have, denoting the hourly increase of mean 
sun's R. A. by .s-. R. A. of mean sun = rt', + 5(P4-A). 

Therefore by the preceding formula. 

P=[a + m (P -h A)] -[«', + s (P + A)J. 
Solving. 

a-a,-\-X{m — s) 
l-{m- s) ' 

In this equation a and o's are given directly in the Ephemeris, 
A is supposed to be known, and s is constant and equal to 
9.8565 seconds: in is obtained from the column adjacent to 
the one giving value of «'. and should be taken so that its 
value will denote the change at the middle instant between 
the Greenwich mean noon and the instant under discussion, 
viz.. i(P+ A), as near as can be determined. 

For the moon, whose motion in R. A. is varied, and for an 
inferior planet, a second approximation may be necessary. 
If the planet have a retrograde motion, m becomes negative. 
If the body be a star, in becomes zero. 

If the sidereal time of culmination be required, the above 
formula holds, substituting for the mean sun the vernal 
equinox, whose R. A. and hourly motion in R. A. are zero. 

Hence, 

_ a -\- \m 
~ 1 — m 

For a star. P'= a. 



92 PRACTICAL ASTRONOMY. 

AZIMUTHS. 

Definitions. In surveys and geodetic operations it often 
becomes necessary to determine the ' ' azimuth " of lines of 
the survey; i. e., the angle between the vertical plane of 
the line and the plane of the true meridian through one of 
its extremities; or, in other words, the true hearing of the 
line. 

For reasons given under the head of Latitude, the geo- 
detic may differ slightly from the astronomical, azimuth of 
a line. Only the latter will be referred to here, and it is 
manifestly the angle at the astronomical zenith included 
between two vertical circles, one coinciding with the me- 
ridian, and the plane of the other containing the line in 
question. 

Outline. In outline, the method consists in measuring 
with the ''Altazimuth" or ''Astronomical Theodolite" the 
azimuth of the line with reference to the verticed circle of 
some celestial body whose place is well known. Then hav- 
ing ascertained by computation the true azimuth of the 
body at the instant of its bisection by the vertical wire, the 
sum of the two will be the true azimuth of the line. 

Instruments. The "Astronomical Theodolite" is provided 
with both horizontal and vertical circles. In geodetic work 
the latter is used largely as a mere finder, but the former is 
often of great size — sometimes three feet in diameter, w4th 
a telescope of equal focal length. For reading the circle, it 
is provided with several reading microscopes fitted with 
micrometers, in lieu of verniers; and in order that any 
angle may be measured with different parts of the circle, 
the latter is susceptible of motion around the vertical axis 
of the instrument. Eccentricity and errors of graduation 
are thus in a measure eliminated. 

To mark the direction of the line at night a bull's-eye 
lantern is ordinarily used. It should be placed not less 
than a mile distant in order to avoid ref ocusing for the star. 

At that distance it will present the appearance of an 
ordinary second magnitude star. By day a target of any 
approved pattern presenting a definite niark, may be used. 



PRACTICAL ASTRONOMY. 93 

Classification of Azimuths. Azimuths of the line with 
reference to the star are taken in "sets,*' the number of 
measurements of the angle in each set being dependent 
upon whether the final result is to be a primary or secondary 
azimuth. Primary azimuths are employed in determining 
the direction of certain lines connected with the funda- 
mental or primary triangulation of a survey, and each set 
consists of from -4 to 6 measurements of the angle in each 
position of the instrument. The final result is required to 
depend upon several sets, with stars in different positions 
(generally not less than five, and often many more). The 
error of the chronometer (required in the reductions), to- 
gether with its rate, are determined by very careful time 
observations with a transit. 

Secondary azimuths are employed in determining the 
direction of certain lines connected with the secondary or 
tertiary triangles of a survey. The number of measure- 
ments in a set is about one-half or one-third that in a set for 
a primary azimuth; the number of sets is also reduced, and 
the time observations are usually made with a sextant. The 
sun is used in connection with secondary azimuths only. 

Selection of Stars. In order to determine the best stars 
to use, let the parts of the Astronomical Triangle be denoted 
as follows: 

2= zenith distance of star. 
()'= declination of star. 
(^=latitude of place. 
P= hour-angle of star. 
^= azimuth of star from the north. 
'/;= parallactic angle of star, or angle at the star. 

An application of the elementary formulas of Spherical 
Trigonometry to the triangle, gives 

tan A = — ^^^^ = , (136) 

cos q) tan o — sm cp cos P 

from which it is seen that with a close circumpolar star, 
small errors in the assumed value of P will have but a very 
slight effect upon the resulting value of A, while errors in 



94 PRACTICAL ASTRONOMY. 

S and cp will have considerable effect. If the circumpolar 
star be taken at its greatest elongation (// =90°,) we have 

( cos d 

sni Ae= , • (137) 

cos^ 

from which it is seen that errors in P will have no effect, 
while errors in ^ or cp will attain their maximum effect. 
But if observations be made upon the star at both eastern and 
western elongations, the resulting true azimuth of the line 
will be independent of these errors, since the angle between 
the line and the vertical plane of the star will be too large 
in one case and too small in the other. In general, effects 
of errors in declination and latitude disappear in the mean 
result of opposite and equal azimuths. Circumpolar stars 
at their elongations (both) are most favorably situated, 
therefore, for the determination of azimuths; and since 
experience gives a decided preference to stars in these 
positions, other cases will not be considered, except to 
remark that the Astronomical Triangle then ceases to be 
right angled. 

The stars a (Polaris), S, and A, Ursse Minoris, and 51 
Cephei, are those almost exclusively used (although the 
latter two cannot be used with small instruments). Their 
places are given in a special table of the Ephemeris, pp. 
302-313, for every day in the year, and they are so dis- 
tributed around the pole that one or more will usually be 
available for observation at some convenient hour. Of 
these four, A Ursee Minoris is both the smallest and nearest 
to the pole. For the large instruments it therefore presents 
a finer and steadier object than any of the others. For the 
small instruments, suitable stars may be selected from the 
Ephemeris. 

Measurements of Angles with Altazimuth. In order to 
understand the measurement of the difference of azimuth 
of two points at unequal altitudes, let us suppose that the 
horizontal circle of the ''Altazimuth" has its graduations 
increasing to the right (or like those of a watch face), and 
that absolute azimuths are reckoned from the north point 



PRACTICAL ASTRONOMY. 95 

through the east to o60°, the orig'in of the graduation being 
at the point 0. Figure 0. 

The angle X-180-'^70-O will then be the absolute azimuth 
of the origin of graduations == O. and if the instrument be in 
adjustment and ^4... and Ai denote the absolute azimuths of 
ihe star and line respectively, we shall have 






in which B and B' denote 



angles OLS and o L L' respectively, and may be con- 
.•<idered as the readings of the instrument when pointed upon 
the star and over the line. These equations w^ll be some- 
what modified if the instrument be not in perfect adjust- 
ment. This will usually be the case. Let us suppose that 
the end of the telescope axis to the observer's left is elevated 
so that the axis has an inclination of h seconds of arc. 
Then if the telescope be horizontal and pointing in the 
direction L S, it will, wlien moved in altitude, sweep to the 
right of the star, and the whole instrument must be moved 
toward O to bring the line of collimation on the star. The 
i-eading of the instrument will thus be diminished to r. and 
we shall have the proper reading. B—r + di correction. The 
amount of this correction is readily seen, from the small 
I'ight-angled spherical triangle involved (of which the 
required distance is the base), to be h cot z. In the same way 
it is seen from the principles explained under ''Equatorial 
Intervals," etc., that if the middle wire be to the left of the 
line of collimation by r seconds of arc. /• must receive the 
correction c cosec z. Hence when both these errors exist 
together, we shall have 

As= + r + b cot 2 + c cosec z. (138) 

Ai= O + r' + b' cot z' +c cosec z\ (139) 

since c remains unchanged, while b is subject to changes. 
Subtracting, 

.4;— ^,= (r' + 6'cot2') — (/* + &cotz)+c(cosec2'— cosecz.) (140) 

Since by reversing the instrument the sign of c is changed 
but not altered numerically, w^e may, if an equal number of 
readings in the two positions be taken, drop the last term 



96 PRACTICAL ASTRONOMY. 

as being eliminated in the mean result. With this under- 
standing, the equation will be 

Ai-A,= (r' + b' cot z') - (/• + b cot z) , (141) 

which gives the azimuth of the line with reference to the 
star, free from all instrumental errors, b is positive when 
the left end is higher, and its value, heretofore explained, 
is obtained by direct and reversed readings of both ends of 



being- the 



"^is 



the bubble, and is — - j (iv + ic') — (e + e') 

value of one division in seconds of arc. For stars at, or very 
near, elongation, it is evident that cot z may be replaced by 
tan cp, without material error, c is positive when middle 
wire is to the left of its proper position. 

For very precise work, the above result requires a small 
correction for diurnal aberration, the effect of which is to 
displace (apparently) a star toward the east point. For 
stars at elongation, this correction is O.^'oll cos A^. 

In using the reading microscopes, care should be taken to 
correct for "error of runs." When a microscope is in per- 
fect adjustment, a whole number of turns of the micrometer 
screw carries the wire exactly over the space between two 
consecutive graduations of the circle. Due to changes of 
temperature, etc., the distance. between the micrometer and 
circle may change, thus altering the size of the image of a 
"space." The excess of a circle division over a whole num- 
ber of turns is called the "Error of Euns." This error is 
determined bv trial, and a proportional part applied to all 
readings of minutes and seconds made with the microscope. 

Observations and Preliminary Computations. The obser- 
vations and the preliminary computations, are as follows: 
The error and rate of the chronometer, error of runs of the 
micrometers, coUimation error and latitude, are supposed 
to have been obtained with considerable accuracy. The 
apparent R. A. and declination for the time of elongation, 
of the star to be used, must be taken from the Ephemeris, 
or if not given there, reduced from the mean places given 
in the catalogue employed, as explained under Zenith 
Telescope. 



PRACTICAL ASTRONOMY. 97 

tan cp 



Then for the star's hour-angle at elongation, cos Pe 
" azimuth '' '' cos .4, 



tan d' 
cos 6 
cos cp' 



'' " '' zenith distance at '' cos z^— -; — ^. 

sm o 

" - sidereal time '' '' 7:=a'±P,. 

'* chronometer '' '• " T,= T„^E, 

n' being the R. A. . and E the chronometer error. 

The instrument is then placed accurately over the station 
and leveled, so that everything will be in readiness to begin 
observations at about '^Cr before the time of elongation as 
above computed. In the actual measurement of the angle 
several different methods have been followed. First, five or 
six pointings are made on the target, and for each pointing, 
the circle and all the microscopes are read; also if the angle 
of elevation of the target differ sensibly from zero (as would 
not usually be the case with the base line of a survey) read- 
ings of the level, both direct and reversed, are made. If 
the target be on the same level as the instrument, cot z' 
will be zero, and the level correction will disappear. Then 
five or six pointings are made on the star, and in addition 
to the above readings, the chronometer time of each bisec- 
tion is noted. The instrument is then reversed, to eliminate 
error of collimation, and the above operations repeated 
beginning with the star. In the second method, alternate 
readings are made on the mark and star, star and mark, 
until five or six measurements of the angle have been made, 
the chronometer being read at each bisection of the star: 
the circle, microscopes and level as before. The instrument 
is then reversed, and the same operations repeated in the 
reverse order. The middle of the time occupied by the 
whole set should correspond very nearly to the time of 
elongation. Similar observations are then made, on the 
same or following nights, on other stars, combining both 
eastern and western elongations, and using different parts 
of the horizontal circle for the measurement. 

Reduction of Observations. Since the observations on the 
star have been made at different times, and since these 



98 PRACTICAL ASTRONOMY. 

correspond to different though nearly equal azimuths, the 
first step in the reduction is to ascertain what each reading 
on the star would have been, had the observation been 
made exactly at elongation. For this purpose find the 
difference between the chronometer time of each observation 
and the chronometer time of elongation as computed, apply- 
ing the rate if perceptible. Let the sidereal interval between 
these two epochs be denoted by r seconds. Then the elon- 
gation reading of the star would have been 

actual reading ± the expression 2(450r)2 g^j^ ^/r ^^^ j^^ 
which denote by C. 

[The quantity 2(450r)^ sin 1" is almost exactly equal to the 
tabulated values of "m'^ in the "Reduction to the Merid- 
ian/' and may if desired be taken directly from those 
tables.] With a circle graduated as assumed, this correc- 
tion would manifestly be negative for a western, and posi- 
tive for an eastern, elongation. Hence Eq. (141) becomes, 
Ai-Ae={r' + b' cot z')-(r-hb cot z±C). (142) 

Each pair of observations (on the line and star) with the 
telescope "direct" gives a value of Ai—A^. If n^ be the 

2{A —A ) 
number of such pairs, the mean will be — ^^ — ^ — ^-^ , to which 

if J.e(positive for eastern, negative for western, elongations,) 

be added as heretofore computed, (sin J-e= ), we shall 

have the true hearing of the line, for instrument ''direct.'' 
Similarly for instrument "reversed," we shall have 

^{A,-A,) 

from which by adding Ae we obtain tlie true heaiHng of the 
line for instrument reversed. 

The mean of the two is the true hearing of the line as 
given by the star employed. 

[For the greatest precision, this must be corrected by 
adding the diurnal aberration, 0".311 cos A^.'] 

The above is the accurate method of reduction. It may 
be simplified, though made less precise, by treating the 
mean of the observations on the line and the mean of those 



PRACTICAL ASTRONOMY. 99 

on the star (instrument * * direct ") as a single observation 
taken at the mean of the times, and reducing it to elonga- 
tion by computing C for the mean value of r. 

Similarly for instrument "reversed." 

This mode of reduction is specially applicable when the 
observations have been made by the first method. 

The adopted value of the azimuth of the line should rest 
upon at least five such determinations. 

DECLINATION OF THE MAGNETIC NEEDLE. 

The Declination of the Magnetic Needle may be found 
in accordance with the same principles, regarding the 
magnetic meridian pointed out by the needle, as the line 
whose azimuth is to be found. Or, note the reading of the 
needle when the instrument carrying it is pointed accu- 
rately along a line whose true bearing or azimuth is known. 
Or, take the magnetic bearing of some known celestial body, 
and note the time T. Then P=T-a. This value of P in 
Eq. (13G) gives the true azimuth, and the difference between 
this and the magnetic bearing gives the declination of the 
needle. 

SOLAR ECLIPSE. 

A solar eclipse can only occur at conjunction, that is at 
new moon, and then only when the moon is near enough to 
the plane of the ecliptic as to encroach upon the umbra or 
penumbra of the cone of light. The following discussion, 
abbreviated from that found in Chauvenet's Practical 
Astronomy, Vol. I, will suffice to give the student such a 
knowledge of the theory of eclipses, as to enable him to 
project a solar eclipse, with the aid of the eclipse data found 
in the Ephemeris. 

Solar Ecliptic Limits.— Let XS Fig. 10 be the Ecliptic, 
NM the intersection of the plane of the moon's orbit with 
the celestial sphere, *N the moon's node, S and M the sun's 
and moon's center at conjunction, and S' and J/' the same 
points at the instant of nearest angular distance of the moon 
from the sun. Assume the following notation, viz. :— 



100 PRACTICAL ASTRONOMY. 

ft = 8M, the moon's latitude at conjunction. 
i = SNM, the inclination of the moon's orbit to the ecliptic, 
A = the quotient of the moon's mean hourly motion in longi- 
tude at conjunction, divided by that of the sun. 
i\ — S'M\ the least true distance. 
y=8M8\ 

Considering NM8 as a plane triangle, and drawing the 
perpendicular M' P from Jf ' to 8N, we have 

88' =/3tsinr. 8P = \ fiisiny. 

8'P= ft {X-l)tsiJir. M'P=ft-\ ft tan y tan ?. 

A ^ = /5^ [(A - 1)2 tanV + (1 - A tan i tan yf']. 

Differentiating the last equation and placing —^ = 0, we 

find A will be a minimum for 

A tan i 
tan y 

This value gives 



or 



{X-iy + XHanH' 


ft^x-iy 


(A - ly + A^ tan^f ' 


^nmal to tan / 



(u:3) 
(144) 



when tan i' is placed equal to ^ .^.. . . 

The least apparent distance of the sun's and moon's center 
as viewed from the surface of the earth may be less than a 
by the difference of the horizontal parallaxes of the two 
bodies. Call this distance a ', then 

A'=A-(P-7r). 
Kow when A ' is less than the sum of the apparent semi- 
diameters of the sun and moon there will be an eclipse; 
hence the condition is (denoting the semi-diameters of the 
moon and sun respectively by s' and s), 

A-{P-7t)<S + s', 

or 

ft cost' <P—7r + s + s\ (145) 

To ascertain the probability of an eclipse, it is generally 
sufficient to substitute the mean values of the quantities in 



PRACTICAL ASTRONOMY. 101 



the above inequality, 
observation are 


The extreme values, 


determined by 


■ j 5° 20' 06" 

^' i 4° 57' 22" 

5° 8' 44" 


p J 61' 32" 
^ ( 52' 50" 

57' 11" 


TT 


j 9". 

( 8".70 

8". 85 


{ 16' 18" 
'' ) 15' 45" 


, j 16' 46" 
•^ } 14' 24" 


X 


J 16.19 
( 10.89 



16' 1" 15' 35" 13. 5 

The mean value of sec i', found from those of i and A, is 

1.00472 and hence, (146) 

/i < (P- ;r + 6' + 6-') sec /' = /?< (P- TT + 5 + 5') (1+0.00472). 

The fractional part of the second member of the inequality 
varies between 20" and 30"; taking its mean 25", we have 
for all but exceptional cases, 

/i < P - TT + 6' + s' + 25". (147) 

Substituting in this last form, the greatest values of P, s, 
and s', and the least value of tt; and then the least values 
of P, s, and s', and the greatest value of tt, we have 

/?<1° 34' 27". 3, 
and 

/3<1° 22' 50", 
respectively. 

If, therefore, the moon's latitude at conjunction be greater 
than 1° 34' 27". 3 a solar eclipse is impossible; if less than 
1° 22' 50" it is certain; if between these values it is doubtful. 
To ascertain whether there will be one or not in the latter 
case, substitute the actual values of the date of P, tt, s and 
.9' and if the inequality subsists there will be an eclipse, 
otherwise not. 

PROJECTION OF A SOLAR ECLIPSE. 

1. To find the Radius of the Shadow on any Plane per- 
pendicular to the Axis of the Shadow. 

In Fig. 11 let aS and M, be the centers of the sun and moon; 
Fthe vertex of the umbral or penumbral cone; FE the funda- 
mental plane through the earth s center perpendicular to the 



102 PRACTICAL ASTRONOMY. 

axis of the shadow; and CD the parallel plane through the 
observer's position. It is required to find the value of CD 
at the beginning or ending of an eclipse. 

Take the earth's mean distance from the sun to be unity 

and let E8=r, EM=r', M8=r-r\ Place *^^ ^ g and 

let k be the ratio of the earth's equatorial radius to the moon's 
radius =0.27227. Then n^ being the sun's mean horizontal 
parallax, we have 

Earth's radius = sin tto. 

Moon's radius = k sin n^ = 0. 27227 sin n^. 

Sun's radius = sin s. 

s being the apparent semi-diameter of the sun at mean 
distance. From the figure we have 

. -^ T^ T-T • ^ sin s ± k sin tt^ 
sm F VE = sm / ^ '- , (148) 

in which the upper sign corresponds to the penumbral and 
the lower to the umbral cone. . The numerator of the second 
member is constant, and since s = 959". 758, 7ro = 8".85 we 
have 

log [sin s + k sin n^ = 7.6688033 for exterior contact, 
log [sin s — k sin zTo] = 7.6666913 for interior contact. 

If the equatorial radius of the earth be taken as unity, we 
have 

VM= ~, MF=z. 
sm/ 

Whence the distance c of the vertex of the cone from the 
fundamental plane is 

k 

c — z±— — ^. (149) 

sm/ ^^^^' 

If / and L be radius of the shadow on the fundamental 
and on the observer's plane respectively and C be their 
distance apart we have 

I = c tan f=z tan / ± A: sec /. (150) 

i = (c - C) tan /= / - C tan /. (151) 



PRACTICAL ASTRONOMY. 103 

2. To find the Distance of the Observer at a given time from 
the Axis of the Shadow in terms of his Co-ordinates and 
those of the Moon's Center, referred to the Earth's Cen- 
ter as an Origin. 

Let 0, Figure 12, be the earth's center, and XY the 
fundamental plane. Take Z Fto be the plane of the decli- 
nation circle passing through the point Z in which the axis 
of the moon's shadow pierces the celestial sphere: XZ be- 
ing perpendicular to the other two coordinate planes. Let 
M and S be the centers of the moon and sun, il/', aS", their 
geocentric places on the celestial sphere, M^ their projections 
on the fundamental plane, and C\ the projection of the 
observer's place on the same plane. Let P be the north 
pole. The axis OZ being always parallel. to the axis of the 
shadow will pierce the celestial sphere, in the same point 
as S M. Assume the following notation: 

a, 6. r =:the R. A., Dec and distance from the earth's 

center, respectively, of the moon's center. 
a\ 6\ r'=:the corresponding coordinates of the sun's center. 
a, d, =the R. A. and Dec. of the point Z. 
.r. y, z =the coordinates of the moon's center, 
if, ;/, C =the coordinates of the observer's position. 
(p, qj' =the latitude and reduced latitude respectively. 
A =the longitude of the observer's station west from 

Greenwich. 
/J =the earth's radius at the observer's station in 

terms of the earth's equatorial radius taken as 

unity. 
>M —the Greenwich hour angle of the point Z. 

^^ =the sidereal time at which the point Z has the 

R. A. a. 
A =the required distance of the place of observation 

from the axis of the shadow at the time yw. 

From the conditions, we have 

R. A. of Z =a, 
R. A. otM' = a 
R. A. of X ==90° + a 



104 PRACTICAL ASTRONOMY. 

and therefore 

ZPM'=a-a, andPM' = 90°-S. 

Through M, and C, draw M^ N and C, N parallel to the 
axis of X and F respectively; then M, C, N^PZM'^P, the 
position angle of the point of contact and we have 

A sin P=x—B,, 

A cos P=^y—f]. (^^^) 

From the spherical triangles Jf' PX, ilf P F. and M' PZ. 
we have 

x'=rcos iiPX[=r cos ^sin («'— a) ^ 

^rnr cos i)f' F=r [sin (^ cose?— cos (^ sin d cos (<a'— a)] I (153) 
2=:rcos JIP^— r[sin (^ sin d— cos (^ cos d cos («'—«)]. j 
Similarly the coordinates of the place of observation are 
^—p cos cp' sin {M—a) -] 

7;=:p[sin ^' cosd— cos ^' sin dcos (yw — a)J [ ^54) 

5=p[sin <7?' sin d + cos <^' cos d cos (yu — a)]. J 

The hour angle {M^—ct) of the point Z for the meridian of 
the observer can be found from 

in which pi^ is the hour angle of the point Z for the Green- 
wich meridian and A is the longitude of the observer's 
meridian. 

The distance of the observer from the axis of the moon's 
shadow A , = C^ 31, can be found from the above formulas, 
since, A'=(x-4')'+(i/-7/)'. (155) 

3. To find the Time of Beginning or Ending of the Eclipse at 
the Place of Observation. 

For the assumed Greenwich mean time of computation 
take from the Besselian table of elements given in the 
Ephemeris for each eclipse the values of sin d, cos d. and pi. 
The values of p cos ^' and p sin ^' are found on page 505, 
computed from the formulas, 

a cos (p 
pcos^'= , ^ . = =PC0S99 

, - (156) 

- sin a>vl— e^sin^<z? sin cp 
p sm ^'= ^-— 27 ~ = —PT^' 



PRACTICAL ASTRONOMY. lOo 

The variations of B, and ?; in one minute of mean time are 
obtained by differentiating the first two of Eqs. (154) and give 
^'= [7.r;309->] fj cos cp' cos (/.^— A) | 

;/ = [7.63992] pcos cp' ^\\\d sin (//-A) \ (15-^ 

= [7. 03992] ^sin d. \ 

The variations of :r and y for one minute of mean time are 
represented by x\ and y\ and their logarithms are given in 
the lower table of the Ephemeris elements for the eclipse. 
Now. if the time chosen for computation be exactly the 
instant of beginning or ending of the eclipse, then i\ — L\ 
but as this is scarcely possible a correction r in minutes 
must be made to the assumed Ephemeris time T. 
We may then write, 

L sin P=.r-ce+ (.r'-c^') r. (158) 

L cos F=ij — i]^ {i}' — if) r. (159) 

Assume, the auxiliary quantities m, 31. h. X. given by the 
equations. 



(160) 



/// sin J/=.r — B. 


1 


m cos M — If — J/. 


1 


u sin X^y -B'. 


1 


n cos .V= //' — ;/. 




From these, we have 




L sin (P-X) = in sin (M - 


^V), 


L cos (P- X) = m cos (J/- 


■ -V) 4 


Hence putting i/- = P— X. we have 




. , HI sin {31— X) 





}} r. 



ni COS (M—X) , Zy cos ?/' 



(161) 



(162) 



(163) 



n n 

the lower sign of the second term in the second member of 
the last equation corresponding to the time of beginning and 
the upper to the time of ending of the eclipse. * 

4. The Position Angle of the Point of Contact.— The angle 
required is P= X+ ?/' for the end and P=.Y-^:±180° 
for the beginning of the eclipse. 

* See Page 0O6. Epliemeris. 



lOf) PRACTICAL ASTRONOMY. 

5. We now have all the equations, and the Ephemeris gives 
us the Besselian table of elements from which the 
circumstances of an eclipse can be computed at any 
place. These equations are here arranged in the order 
in which they would.be used, and the student is referred 
to the type problem worked out in the Ephemeris as 
a guide. 

1. Constants for the given place, 

p sin Gp' ] Found from table page 505. Ephemeris, 
p cos q)' \ knowing the observer's latitude. 

2. Coordinates of observer, referred to center of earth, 

B — p cos (p' sin (yw — A). 

7/ = p sin q)' cos r/ — p cos q/ sin d cos (// — A), 

C = p sin <// sin c/ + p cos qj' cos d cos (p — A). 

3. Variations of observers coordinates in one minute of 

mean time, 

B,' = [?.()3r)92] p cos q)' cos (p - A). 
jf = [7.63992] <? sin d. 

4. The values of m, M, n and N, given by 

m sin Jf = .T — ^. 
m cos M=y —ij, 
n sin N = x' — ^' . 
n cos N=t/ — 7]'. 

5. The radius L of the shadow or penumbra on a plane 

passing through the observer parallel to the funda- 
mental plane, and at a distance Z from it, 
L = l-C tan /'. 
0. The value of the angle ip, 

. , m sin (M—N) 
sm f = Y • 

7. The value of the time r in minutes 

_ m cos (M—N) L cos tp 
~~ n ~~ n ^ 

8. The position angle P, from 

or 




Fig. 3 



E H 

Fig. 4. 





Fig. 6a. 



Fn. 8. 




S N 




Fig. 9. 




Fig. 12. 



H 
m 






z 






o 

I— ! 






o 

a 

o 
o 

O 



o 
^ 



OS 



o 

o 
p^ 

P:h 



I— I 
o 
Ph 

H 



O 

•rH 









-(-= 








o 








® 








f^ 




. 


. 


u 




u 


Jh 


o 




CD 
> 




O 




;h 


?H 


1— 1 


q:- 


o 


o 


a; 


H^ 


OQ 


o 


!> 


03 


rO 








P 


O 


W 


f-l 



II 

o 



jS 



o .s 



o 
O 



o 


O 

O 

o 
•1—1 
+=> 

s 



o 

o 



s .^ 



^' ? 



o 



o 


o 
O 



c5 



o 
Q 



02 



p4 

Ph 
Ph 



N 



O 

O 



o 



F^ORJVC Islo. 1. 

ERROK OF SIDEREAL TIME-PIECE BY MERIDIAN TRANSIT OF STAR. 
Station, West Point, N". Y. Latitude, 41° 23' 83". 11. Chronometer No , by 



Date: 

Observer. 

Recorder. 

Transit. 

Illumination. 

Name of Star. 

f Direct. 
Level. J 

j Reversed. 


E. W. 

E. W. 


E. W. 
E. W. 


E. W. 
E. W. 


E. W. 
E. W. 


E. W. 
E. W. 


i 

H 


I. 
11. 
III. 
IV. 
V. 
VL 
l^VIL 


h m s 


h m s 


h m s 


h m s 


h m s 


Sum. 












Mean. 

Red. to Mid. Wire. 












Chron. Time oi Transit ] 
over Mid. Wire^T j 
Level Error =6. 
Level Correction =£&. 
CoUimation Error =--c. 
CoUimation Correction =Cc. 
Azimuth Error = a. 
Azimuth Correction =J.a. 












Chron. Time of Transit. 
App. R. A. of Star = a. 












E 


rror of Chron. =J5. 













(M 



O 



ERROR OF MEAN-SOLAR TIME-PIECE BY MERIDIAN TRANSIT OP SUN. 

■Date. Station, West Point, N. Y. 

Latitude 41°-23'-23".ll. Longitude 4.93*'. 

Observer. Recorder. 

Transit No By Mean-Solar Chronom. No By . . 



Chronometer time of Transit of West Limb. 



Chronometer time of Transit of East Limb. 



Wire 1 

" II 

" III 

" IV 

" V 

" VI 

" VII 

" I 

" II 

" III 

" IV 

" V 

" VI 

" VII 



SUM. 
Chron. Time of Transitof Center over Mean of Wires = Mean 
Reduction to Middle Wire. 

Level Error Level Correction. 

Col. " Col. 

Azimuth " Azimuth " 

Chronom. Time of App. Noon. 

Apparent " " " 

Eq. of Time. 

Mean Time of Apparent Noon. 

Error of Chronometer on Mean Solar Time at App. Noon. 



.12^ 



cos a cot a J ^ 
T being the different clironometer times. The last factor is taken from Tables as before. 



F^OI^IVl No. 3. 

ERROR. OF SIDEREAL TIME-PIECE BY SINOLE ALTITUDIQ OF H'I'AR. Ny\ME 

p.,(^. Station, Wkht Point, N. Y. 

cn)server. Recorder. 

Sextant No By Sidereal Chronom. No By 

" ' " h m s 

Observed Double Altitude. Chronom. Time. 



Index Error. 

Eccentricity. 

Corrected Double Altitude. 

" Altitude = a,,. 

♦Refraction =r. 

True Altitude = a. 
Latitude =9?. 
N. Polar Dist.=d 
a+qf + d 



,41°.. 23'.. 23". 11. 



Sum 
^Mean=:^<, 



Barometer 
Att. Thermom 
Ext. 
Refraction 



a. c. log cos qj 

" " sin d 
" cos in 
" sin(7H— a) 

" sin i P 
iP 
P 
P in Time 
Apparent R. A. of Star 
Sidereal Time =R. A. +P 
Mean of Chron. Times = ifo 
Error of Chronometer 



*The correction to be added to this value of r, if 



Note 3, Text), is 2 -^ 



- ffl(, n sin 1 

taken from Tables (first converting n^ — A into its equivalent in time). 



a^ their mean, and n the number of observations. The values of 
as explained under " Latitude by Circum-Meridian Altitudes." 

**The correction to be added algebraically to this value of t^ if desired (See Note 3, Text) is, after computing p in arc, 

r ^_,sinPcos^sinc^-| ^ ^ ^.'- UT-Q 
' ^ L cos a cot a J ii sin 1 " 

T being the different chronometer times. The last factor is taken from Tables as before. 



to 



^ 



o 



^ 



F^ORJM No. 4. 

ERROR OF MEAN-SOLAR TIME-PIECE BY SINGLE ALTITUDIO OF SUN'S LIMB. 

J)^^^ station, Wicst Point, N. Y. 

Observer. Recorder, 

Sextant No By M. S. Chronom. No By 

Observed Double Altitude. Chronom. Time. 



Mean " " 

Index Error. 

Eccentricity. 

Corrected Double Altitude. 

" Altitude =«„. 
*Refraction=r. 
Semi-diameter. 
Apparent Altitude = a'. 
Parallax in Altitude. 
True Altitude = a. 
Latitude = (p. 
N. Polar Dist.=d 
a+q) + d 



.41°..23'..22".ll.... 



Sum 
**Mean=4 

Barometer 
Att. Thermom 
Ext. 
Refraction 

Longitude = 4.931 hours. 

Assumed Error of Chronom. = 
Resulting Greenwich Time of Obs. 

Log Eq. Hor. Parallax 

" p. 

" cos a'. 
Parallax in Altitude. 

Dec. at Greenwich Mean Moon. 
Hourly Change x Greenwich Time 
Sun's Declination. 

a. c. log cos cp 

" " sin d 

" cos m 

" sin {m—a) 

" sin^P 

P 

P in Time 

Apparent Time 

Equation of Time 

Mean Time. 

Mean of Chron. Times =^o 

Error of Chronometer 



*See Foot Note to Form 3. 
#» << <( .< .< <i .. 

Note. — For correction of Semi-diameter due to difference of refraction between limb and center, 



'Longitude by Lunar Distances. 



ERROR OF SIDEREAL TIME-PIECE BY EQUAL ALTITUDES OF A STAR. 
Station, West Point, K Y. Latitude, 41° 23' 22". 11 = cp. 

Observer, •. Recorder. 

Sextant No By Sid. Chronom. No By 

Name of Star App. Declination =d 



I. 
II. 
III. 



Observations East. 
Observed Double Altitudes. 



Chronometer Times. 
h m s 



Date 



Bai'om. 

Att. Thermom . 

Ext. 

1st Refraction . 



Mean — 2a 



(Correct this for index error, if correction for 
refraction be taken into account). 



Observations West. 
Observed Double Altitudes. 



Sum_ 
1st Mean. 



I. 
II. 
III. 



Mean = 2 a . 



Same as above). 



Elapsed Time. 

i Elapsed Time in arc = t. 



Chronometer Times. 
/), m s 



Date. 



Barom. 

Att. Thermom . 

Ext. 

2nd Refraction . 

.1st 



Middle Chronomotor Time. 

Correction for Refraction. 

Chrononi. Time of Transit. 

App. R. A. of Star. 

Error of Chronom. at Time of Transit 



. . Sum Difference 

2d Mean Log Difference . 

1st " . Log cos o 

a. c. log ;iO 

.' a. c. log cos q> . 

a. c. log cos 6 . 

a. c. log sin t . 

Log Correction . 

Correction 



NOTK. — For Tlioory of Time by Etjual .\ltitiides, see Young, pp. 77 — 78. If the refractions are different at the two obserration.s, the true altitadea 
win be uuetiual when the observed are equal. The correction for refraction compensates for this. To avoid the corrections referred to in Foot Note 
Form 8, not more than 8 double altitudes should constitute a •' group." The groups may be multiplied at pleasure. 

luglcs. This is corrected by the " Equation of Equal Altituaes. 



p 



FORiVt 6. 

ERROR OF MEAX-SOLAR TIME-PIECE BY EQUAL ALTITUDES OF SUN'S LIMB. 

Station, West Point, N. Y. 9> = Latitude, 41° 23' 22". 11. Longitude 4. 9ol*, west. 

Observer, Recorder 

Sextant No By M. S. Chronom. No By 

Sun's App. Dec. at local App. Noon (or midnight) = 6= Hourly change in S at same time, =k= ... 



I. 
II. 
III. 



Observations East. 
Observed Double Altitudes. 



Chronometer Times. 
h m s 



Date 



Barom. 

Att. Thermom . 

Ext. 

1st Refraction . 



Mean = 2 a 



(Corn^ct this for iiuUtx error, if correction for 
refraction be taken into account). 

Observations West. 
Observed Double Altitudes. 



Sum. 
1st Mean. 



T. 
11. 
III. 



Mean = 2 a . 



Date. 



Chronometer Times. 
h VI s 



Klapsi'd Time. 

A Klapsi'd Tiiiu> in arc = t. 

MiddK-rhr..iioiiu>torTiiiu>. 
(.'oinvtion for Hi>f rait ion. 
Kiiuation of Equal Altitudes. 
I'hronom. Tiino of App. Noon. 
App. of Time App. Noon. 
Eq. of Timo lit App. Noon. 
Mi>nn Titn«> of App. Noon. 
Error of Chronon>»»tor at App. Nt 



Barom. 

Att. Thermom. 

Ext. 

2nd Refraction . 
-1st 
.Dilfercncf 



. . Sum 

2d Mean Log Differenot 

1st " ..Log cos a 

a. c. log :10 

(t. C. log cos (/J 

a. c. log cos d 

a. c. log sin ( 

Log Correction 

Correction 



Log A. 



1st Part + 2d Part : 



1st Part. 
1st Part. 
.Log//. 
•• /:. 
" tan 'V 
• idPui. 
2d Part. 
Ek). of Ekiual Altitudes. 



Not* I. Sw Foot Note to Konu 5, , 

Kot* 9. Thp Aun rhangvs its decllnatioD betwoea th« tlmea of Bast and West obaermtioiM. Equal altitadei <lo not therefore ' 
K1«s. This ts corrwtpd bv the " Gqaation of Bqaa) Altitades." 



f 



LATITUDE BY CIRCUM-MERIDIAN ALTITUDES OP SUN'S LIMB. 

Date Station, West Point, N. Y. Longitude 4.03". Assumed Lat. = 93= 

Observer Recorder Barom Att. Th Ext. Th 

Sextant No By M. S. Chronometer, No By 

Error of Chronometer =-£■= Rate of Chronometer=r= 



Observed Double Altitudes. 


Chronometer Times. 
7i m s 


App. Time of 

App. Noon 12 
Eq. of Time at 

App Noon 


Hour Angles. 
m. s. 


m. j ^. 
« ! ». 


I. 

II. 

III. 










IV. 

V. 

VI 


Mean Time of 
App Noon 




VII. 






VIII. 
IX. 


Chron. Time of 




X. 1 






Sum 


Log. Eq. Hor. Par Sums. 








Mean 


Po 


1)1 


11 


Eccentricitv " cos a* Means. 







Index Error. 
Cor. D. Alt. 

" S. " 
Refraction. 
Semi-diam. 
Par. in Alt. 
True Alt. =a„ 
Ao mo + 
B„ iio - 
Sum — 
90° + 



.90°...0'.0...0".0. 



Par. in Alt. 



Eq. of Time at P„ 

Po 

— Longitude 

Corresponding Greenwich Time 



Sun's Dec. at 



a^ = So + 90°-(p. 



^=(y„-f 90° — a,. 



Change in Eq. of Time in 24" =e 
Rate of Chronometer " =r 



log 



r— e 

A; 

cos 90 

cos (5"„ 

sec a, 

Ao 

Vlo 

Ao mo 
Aomo 



2 log Ao 
" tan a. 



BoHo 

Bono 



*a is obtained by applying refraction and semi-diameter to Corrected Single Altitude. 

Note. For correction to Semi-diameter due to difference of refraction between limb and center, see Longitude 



Lunar Distances. 



KORJM No. 8. 



LATITUDE BY CIRCUM-MERIDIAN ALTITUDES OF (NAME OF STAR) 

Date Station, West Point, N. Y. Assumed Lat. — q)= 

Observer Recorder Barom Att. Ther Ext. Ther . 

Sextant No By Sidereal Chronom. No By 

Error of Chronometer =J5J= Rate of Chronometer =r= 



Observed Double Altitudes. 


Chronometer Times. 
li m H 


App. R. A. 
of Star 


Hour Angles. 
m. s. 


\ m. : n. 
I s s. 


I. 

II. 

Ill, 










IV. 
Y 


Chron Error .... 




VI. 

VII. 


Chron. Time of 
Transit 




VIII. 

IX. 

X. 






Sum. 


Sums. 












Po 


1»o 




Eccentricity 




Means 


no 





Index Error. 
Cor. D. Alt. . , 

" S. " 
Refraction. _ 
True Alt. =ao-l- 
Ao mo + 
Bo n-o - 
Sum — 
90° + 



.90°...0'.0...0".0. 



Star's App. Declination = So 

a^ = do + 90°-(p. 



^0+90° -a. 



Rate of Chronometer in 24'' ■■ 



log 



k 

cos <p 

cos So 

sec a, 

Ao 

Too 
-4oTOo 

Ao nio 



2 log Ao 
" tan a. 



BoUo 

Boflo 



rui iSo. 96, whicli is for the com- 



PROGRAMME FOR ZENITH TELESCOPE. (LATITUDE.) 



Station, West Point, N. Y. 




Approximate 


Latitude 


Observer 






No. 


Catalogue and 
No. 


Mag. 


Mean R. A. 


Mean Dec. 


Zenith Dist. 


N. 
S. 


Setting. 


1. 
2. 
















3. 
4. 
















5. 
6. 
















7. 
8. 
















9. 
10. 












1 





korm: no. 9a. 

OBSERVATIONS WITH ZENITH TELESCOPE. 

Station, West Point, N. Y. Observer Recorder 

Telescope No By Chronometer Error. 



Sheet No. 



Catalogue & N. 
" No. S. 



MICROMETER. 



VU„&?W, m, 






CHRONOM. 



Time. 



Note.— Forni Xo. 9 is for the observer's use. 
Form 9 II is for the recorder's use. 

The n>cords of the different nights at a given station are then collated, and the reductions made as per Fonii No. 9 b, which is for the com- 
VUter's use. 



St 



FORM No. 9 b. 

EEDUCTION OF OBSERVATIONS WITH ZEmTB. TELESCOPE. 



Station, 



West Point, N. Y. 



Observation Sheet, No. 



Observer . 



Telescope No By . 



One Div. Micrometer {R) ■ 



Recorder Computer . 

... One Div. Level (Z))= 



Date. 



STAR. 



Catalogue & N. 
" No. S. 



MICROMETER. 



m„&m^ m„—m 



In&l'n 



Is&l's 



{L+l'n) 
(Is + l's) 



HOLTR 

ANGLE. 



P. 



DECLINATION. 



Micrometer =-B 



nin-lth 



Level =^ 4 

Note. Corrections. -{ p, dr m„— Ws 

Refraction = godz 3 
^Red.toMer.= [6.1347] P^ sin 2 c^. 



CORRECTIONS. 



Microm. Level. Ref, 



Red. to 
Mer. 



Latitude. 



Columns headed v and vv are for the determination of the prob- 
able error by the "Method of Least Squares. 



Sta 
Se:s 



N 



Nd 



FORM No. lO. 



station, 



LATITUDE BY POLARIS OFF THE MERIDIAN. 

West Point, N. Y. Date Observer Recorder. 

Sextant No By Index Error Sid. Chron. No By 

Barometer Att. Thermom Ext. Thermoni Refraction . . 

App. R.'A. of Polaris App. Dec. of Polaris i-- 



Error . 




Note. The column of True Altitudes is obtained from tlie preceding by apph -ng 
instrumental errors and refraction 
Log. sin 1" =4.68577-10. 



KORTVC No. 11. 

LATITUDE BY EQUAL ALTITUDES OF TWO STARS. 
Station, West Point, N. Y. Observer Recorder 



Date Sextant, No. 

Sid. Chronom. No By 



By. 



Error Rate . 



Na^eoftteStar. ^fc^^ 


True Time of 
Observation. 


App. R. A. 


Hour Angle 
in Time. 


Hour Angle 
in Arc. 
P &P' 


App. Dec. 


d'—S 

3 


S' + d 
2 


P'—P 
2 


P'+P 
2 


1. 
2. 


















1. 
2. 




















1. 

2. 




















1. 
2. 






















1. 
2. 






















Log tan — - — 












Log cot - .^ - 












Log cot ,^ 












Log tan M 
M 




















a. c. log cos M 

P'—P 
Log cos ^-.-^ 






















Log tan - -;^ — 
Log m 




















P'+P 
2 












M 






















Logcos-^tZ_iy 


i i 






Log m 


' 1 






Log tan cp 


! j 






9 


i 1 






Mean 

» — 


i 




1 













I 



stations. 
(1) Eastern. (2) Western 



Long. I Mean. 

V. Signals, y+r' 

\" 2 



Retardation. 

V-\" 



Sum 
Mean, 



FORiVL No. 12. 

LONGITUDE BY ELECTRIC TELEGRAPH. 



Stations. 
(1) Eastern. (2) Western. 



Date. 



Observer. 



No. of 
Time 

Stars. 



Chronom. 

Errors at 
Epoch, 
E&E' 



E-E' 



No. of 
Chronom. 
Compari- 



T-T' 



Dif . of Long. I Mean, 
by E. & W. Signals. A' + A' 
A' & A" 



Retardation. 
A'-A" 



Sum. 
Mean. 



FORM xo. 12 a. 



LONGITUDE BY ELECTRIC TELEGRAPH 
Reduction of Transits of Stars observed at . . 



No. 



.., By. 
h m 



•■• Date ^Vith Transit 

to determine correction to Sidereal Chronometer No Bv 



at. 



Chronom. Time. Assumed Error at Epoch . 



Rate. 



Names 
of Stars. 



Observer. 



Time of Transit 

over 

Middle Wire. 



PI 



Correction, 
for Rate. 



Correction, 
for Level. App. R. A. 



Normal Equations. 

1 

2. ... 



.4/(. 



Cii. 



A C. 



2{7l) 



a 
c 



2{A) 



2{C) 



2{An) 



2{Cn) 



2{AC) 



2{A') 



2{C' 



PRACTICAL ASTRONOMY, 



BY 

P. S. MICHIE 

AND 

F. S. HARLOW. 




U. S. M. A. PRESS SS^^^ y>. AND BINDERY 



1891. 





















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